Tax classes in Germany: TL;DR
In nutshell; your tax class in Germany is more or less defined by your marital status. Depending on this, you will be put in different categories by the Finanzamt. This will help your employer apply the correct tax rate on your payslip, hence the name “Lohnsteuerklasse”. This impacts how much wage tax (“Lohnsteuerabzug”), solidarity contribution (“Solidaritätszuschlag”) and Church tax (“Kirchensteuer”) will be taken from your gross salary.
In other words, your net salary is directly related to your tax class in Germany.

There are 6 categories overall

Tax class	Marital status
Steuerklasse 1	You are single, widow(er), separated/divorced
Steuerklasse 2	You are a single parent
Steuerklasse 3	You are married with a higher income (than your partner in Steuerklasse 5)
Steuerklasse 4	You are married with both equivalent income
Steuerklasse 5	You are married with a lower income (than your partner in Steuerklasse 3)
Steuerklasse 6	The class used for additional jobs (Nebenjob or Zweitjob)

Attention married people: you can save money
As you can see, married people can belong to different classes and optimize differently. By default, the Finanzamt will put both partners in tax class 4, when you first register your marriage or marry in Germany. However, for a lot of households, this is often not the best solution. Why, you ask?

It’s quite simple really: the German tax system knows that the higher-earner is often bearing more of the costs in the household. The idea here is to allow for a tax rebate when one of the partners earns significantly less than the other. As a result, moving from tax class 4 to tax class 3/5 might increase the net salary for the higher earner.