There are a lot of costs that you can claim for in your tax return which will lower your taxable income, i.e. which are deductible. 
Lowering your taxable income means that you will get a part of your pre-paid income tax back. 

Among the deductible costs are:

• your commuting costs, a flat rate of 0.30€ per km for the first 20 km of the simple distance between your home and your workplace (from the 21st km: 0.38€ per km), per day that you worked that year, no matter whether you drove, biked, walked or beamed yourself there, but limited at 4,500€ per year if you weren’t using your own or the company car. Or your real yearly train/public transport costs, with no limit.
• any costs that you had for your job and that weren’t reimbursed by your employer, e.g. to attend a conference (conference fee, travel and extra subsistence costs) or on business trips (most employers don’t pay the maximum per diems allowed by law, especially on trips outside Germany, so you get to claim for the difference), your laptop if you need one for your work and your employer didn’t provide one, the desk and office chair you use when you work from home (no matter whether you have a separate home office that you’re claiming for, or not and therefore only claim the 1,260€ yearly flat rate for home office costs) and so on.
• working from home, in a separate home office: 
  If the main part of your work is performed in your home office, then you can claim the real cost of your home office. If this “real cost” is less than 1,260€ per year, you can instead claim for the 1,260€ (= 210 days * 6€/day) home office flat rate. Please note that you additionally get to claim for your phone/internet costs.

  The home office needs to be a separate room with a door, you need to use it more than 90% for your work and it is not allowed to contain a bed, TV, fitness equipment or wardrobe. It is not allowed to be a room you have to go through to get to another part of the flat (except if that other part of the flat is the bedroom or bathroom).

  You can claim for a share of your “warm” rent (which usually includes nearly all utility costs), plus of any other related expenses like electricity or gas (if not included in “warm” rent).
  If you have a home office in your own flat/house, then you can claim for a share of the depreciation costs and of the utility costs (electricity, gas, water, sewage, garbage, Grundsteuer=real estate tax, and so on).
  Please note that you additionally get to claim for your phone/internet costs.

  A note of caution: if you are self-employed and own this flat/house, working from home will turn that room from “private asset” to “business asset”, which means that when you stop working from home, e.g. because you stop working in that room because you move your business into bigger rented offices or because you sell your home, the Finanzamt will come and make you tax the increase in value of that room as income, at your own personal income tax rate of up to 42%. So if you live in a region with rising real estate prices, you may want to put a guest bed into your office, just to make it ineligible as a home office and to not have the Finanzamt ask for its share in the increase in value of your home. Of course, this means foregoing claiming for the home office as a business expense, but in a fast rising real estate market, that may be more advantageous than taxing the increase in value on that home office.

  If you are 55 years or older, there is a special tax-free allowance for giving up/selling your business of 45,000€ in §16 (4) EStG, which you can use when stopping to work from your home office. So if you plan to work to over age 55 and think the increase in value of your home office will only be up to 45,000€, by all means work from home and claim for the costs, even if you work self-employed.

• working from home, you don’t have a separate home office: 
  If you do not have a separate home office, but work, for example, at a desk in your living room, you can still claim a flat rate of 6€ per day, for up to 210 days per year (210 * 6€ = 1,260€) for the days you worked from home.
  Please note that you additionally get to claim for your phone/internet costs.

• your educational costs, e.g. what you spent for your university studies that year or on training courses. If this was your first degree (Bachelors) you can claim for up to 6,000€ education costs per year (Ausbildungskosten), for all further degrees (second Bachelors or a Masters degree) and training courses, there is no limit to what you can claim for (Fortbildungskosten). 
• a part of Steuerberater invoices (= of my invoices).
• 50% of your accident insurance, as laid down in chapter 1.3 of BMF-Schreiben v. 28.10.2009 – IV C 5 – S 2332/09/10004 BStBl 2009 I S. 1275 
• moving costs if you had to move to take up this job. You can claim both the real costs you had (travel costs, hotel cost when flat hunting, transport of your furniture, overlapping rental costs, … ) plus additionally a flat rate (Umzugspauschale) that’s meant to cover all the other miscellaneous moving expenses for which people usually don’t gather receipts. This additional flat rate for miscellaneous moving expenses rises by 50% if this was your second move for your job within the last 5 years.

• the cost of your second home (doppelte Haushaltsführung) in Germany, that you needed because it’s where your job/business is, while you are still paying for the upkeep of your first home in your home country, because your spouse/children still live there.
  You can claim up to 12,000€ a year for the warm cost  (= including utility costs and TV licence fee) of your second home in Germany (rent or depreciation&interest if you bought it). 
  You can also claim for items for furnishing the second home like kitchen cabinets, a stove, a fridge, crockery, cutlery, washing machine, a table, chairs, a bed, lamps, curtains (the idea behind this is that you need extra furniture for the second home since all the “good” furniture is back home, being used by your family). If an item cost more than 952€ including VAT, you cannot claim for its whole cost immediately but have to depreciate it over its lifetime, which for furniture is assumed to be 13 years! So be moderate and buy cheap furniture and appliances for your second home.
  You also get to claim for your trips back home to your spouse/children, if you fly the cost of the flight ticket, in all other cases a flat rate of 0.30€/km times the simple distance between your second and first home, or the real train costs.
  For the first 90 days you can also claim 28€ per day for additional meal costs, i.e. 2,520€ in total – the idea behind that is that during that period you subsist on expensive takeout, until you orient yourself in the new city and learn where to buy food that you cook yourself, or until your wife also moves and again starts cooking for you 😉

  Please note that if you claim for the “cost of your second home (doppelte Haushaltsführung)”, you cannot also claim for the “flat rate for miscellaneous moving expenses (Umzugspauschale)” for your own, first “small” move to Germany, they are mutually exclusive.
  However, you can claim the “flat rate for miscellaneous moving expenses (Umzugspauschale)” for you and your family when your family joins you in Germany, i.e. for the “big” move, when you change the “centre of your life” to Germany (source).
  Some relocation agencies have the bad habit of incorrectly registering the entire family in Germany on the trip when the whole family visits Germany just to choose the flat/house, even though in reality, one spouse will stay behind with the children and let them finish the school year in their home country – sadly, this means that in the eyes of the Finanzamt, all of you moved to Germany at the same time and that you are therefore no longer eligible for “doppelte Haushaltsführung”.

• health costs not covered by insurance (außergewöhnliche Belastungen), but depending on your total family gross income, your marital status and how many children you have, only costs exceeding 1% to 7% (zumutbare Belastung = bearable burden) of your total family gross income will have an effect. So for most people, the only time they will benefit from this is in a year in which they had very high dental expenses not reimbursed by insurance.
• financial support that you pay to an indigent parent over 65 (if the parent has a doctor’s note that he/she can’t work, also under 65), or to your child under 25 for whom you don’t get Kindergeld, or for your spouse who lives outside Germany, or for an indigent partner (= someone you’re not married to who stays at home in your German household).
  Up to the amount of the Grundfreibetrag (in 2023: 10,908€, in 2024: 11,604€), less if that person lives in a country with a lower cost of living. For each supported person outside Germany, you need to submit a maintenance form that is signed both by the supported person and by that country’s local authorities. 
  Please note that if the supported person does not live in your German household (and if it isn’t your spouse you are supporting), you have to pay the support by bank transfer into the supported person’s bank account and that you need to pay the support in advance, i.e. the worst you could do is transfer them just one big amount in December, since then you would only get to claim for 1/12 x Grundfreibetrag x factor for that country, and the rest of that big amount would not count for the next year. So best transfer the support in January!
• the child tax allowance (Kinderfreibetrag). There are two cases for which you will save tax when using the child tax allowance:
       – you’re a high-earner, which means that you will save more in tax by applying it than you got in Kindergeld
       – you didn’t get any Kindergeld, because you forgot to apply for it or you had the wrong residency permit or your child lives outside Germany. I always need to know about all your children, including the ones that do not live with you in Germany!
• childcare costs for a child under age 14, e.g. for a day nursery, kindergarten or after-school care (Hort). Only 2/3 of the costs will count, and those 2/3 are only allowed to be up to 4,000€ per year per child, so you can only claim for costs of maximum 6,000€ per year per child.
• school fees for your child, for a school in Germany/EU/EEA or for any “Deutsche Schule” worldwide. You can claim for 30% of the school fees, but those 30% are only allowed to be up to 5,000€ per year per child, so you can only claim for costs of maximum 16,666.67€ per year per child.
• you donated money/goods/your time (e.g. by volunteering as a sports coach and refusing the pay/expense allowance) to a German/EU/EEA good cause or you gave money/goods to German political parties.
  Eligible good causes are German/EU/EEA public institutions (e.g. a public university, school, library , hospital or museum) and German/EU/EEA charitable, benevolent or religious organisations that are recognised as such in their country and German political parties. Eligible German political parties are the ones that fulfill §2 Parteiengesetz. 
  Membership fees to such eligible recipients are also deductible, except for membership fees for some clubs among them being sports clubs, cultural/heritage clubs, carnival clubs, veteran clubs and gardening clubs.

  For donations/membership fees up to 300€ that were done by bank transfer, your bank statement is the necessary donation receipt (Zuwendungsbestätigung, also called Spendenquittung), but it has to clearly show an eligible organisation/institution as the recipient and what the donation was for, e.g. you wrote “donation to provide a free breakfast in school for children from low-income families” in the subject line when you transferred that money, or “Mitgliedsbeitrag …-Partei”, see §10b EStG and §50 (4) Nr. 2 EStDV.
  In all other cases, the recipient has to provide you with a receipt for the donation, which in the case of a German recipient has to have the format set down in Anhang 37 Amtliches Einkommensteuer-Handbuch. If it is a non-German recipient, you will have to bring proof that the recipient is eligible, and for the receipt it would be a good idea to use the Anhang 37 Amtliches Einkommensteuer-Handbuch format as a template (note: if a document is not in German, the Finanzamt can ask for a translation before accepting it).

  If you gave up to 20% of your income to a German/EU/EEA good cause, what you gave will reduce your taxable income.
  If you gave more than 20% of your income to a good cause, the amount above 20% of your income will be carried over to the next year and will reduce your taxable income then.

  If you gave to German political parties, you get back 50% of the first up to 1,650€ (if you’re married: 3,300€) as a tax rebate, and the next up 1,650€ (if you’re married: 3,300€) will lower your taxable income.
  Example: you’re married and gave 10,000€ to eligible German political parties. This means that you will get back 1,650€ (= 50% of the first 3,300€) of the income tax that you and your spouse have prepaid. Additionally, your taxable income will be lowered by the next 3,300€. The remaining 3,400€ (10,000€ – 3,300€ – 3,300€) are of no benefit to you, because unlike donations to a good cause, amounts given to political parties that exceed the limit are not carried over to the next year.

• your contributions to German public pension insurance (you’re also allowed to contribute voluntarily, regardless of nationality), or to non-German public pension insurance (e.g. to the UK, if you pay NI contributions to HMRC), or to your eligible US pension plan, e.g. a traditional IRA or one of the other kinds listed in the protocol to the double taxation agreement between Germany and the US (except a Roth-IRA) that you had already paid into before starting to work in Germany, for details please read this thread, or to a special kind of German private pension called Rürup pension which pays out a life-long pension, starting with age 62 at the earliest and can also include occupational disability. 
  
  But they will only have an effect if your total contribution to public, eligible US pension plans and Rürup pensions didn’t exceed a certain limit (Höchstbeitrag zur knappschaftlichen Rentenversicherung) which changes every year. In 2023, this is limit is 26,528€ for a single, and 53,056€ for a married couple (if your spouse doesn’t use their part, you’re allowed to use it).

• the alimony you pay your separated/divorced spouse who lives in Germany/another EU country/a EEA country (Norway, Iceland, Liechtenstein), but only if that (ex-)spouse taxes said alimony as income (Realsplitting). If you are an EU/EEA national, your (ex-)spouse is also allowed to live in Switzerland (H1a “Wohnsitz in der Schweiz” EStH, you need to click on the + to clap open that section). Some double taxation agreements also have an “alimony” clause, so if your ex lives in one of those countries, e.g. USA, Canada or Australia, then you also get to claim it as an expense, but only if your ex in turn taxes it as income.
  You can only claim for up to 13,805€ per year.

Employees automatically get an allowance for job-related costs of 1,230€ (Arbeitnehmer-Pauschbetrag), this is already considered by your employer when calculating how much income tax to deduct from your salary. 
This means that any costs that you had because of your job like commuting costs, conference costs, home office, educational costs related to your present job, moving costs and second home costs will only have an effect if their summed up amount exceeds 1,230€.

Example:

• an employee lives 60km away from his workplace, and commuted to work by train on 110 days from July to December. His train ticket cost 588€.
• he also attended a technical conference because of his job, but his employer did not reimburse him the conference fee, travel and hotel costs and extra subsistence costs of in total 800€. 

His commuting costs are:

• flat rate: (0.30 €/km * 20km + 0.38€/km * 40km) * 110 days = 2,332€, since that’s more than his real train costs of 588€

His job-related conference costs were 800€.

His employer already automatically considered a deduction of 1,230€.
If he now claims for the commuting and conference in his tax return, only:

• (2,332€ + 800€) – 1,230€ already used up allowance = 1,902€ will further lower his taxable income.