When shopping, we pay sales tax ("Umsatzsteuer", colloquially also known as "Mehrwertsteuer"), so actually, we all pay taxes all the time. When the term "taxpayer" ("Steuerzahler") is mentioned, however, it generally has a more confined meaning and refers to people or firms earning an income, i.e. those who make money. Ultimately, everyone who has an income pays income tax ("Einkommenssteuer"). For employees, income tax means payroll tax ("Lohnsteuer"). Larger companies such as Limited Liability Companies (GmbH) or Public Limited Companies (AG) pay corporate income tax ("Körperschaftssteuer").
The other types of taxes which may incur depends on individual circumstances. For instance, dog owners pay dog tax ("Hundesteuer") for their pet, car and motor vehicle owners pay car tax ("KFZ-Steuer") and those who register a trade in Germany, must usually pay trade tax ("Gewerbesteuer"). You can find an overview of different types of tax on the website of the Federal Ministry of Finance ("Bundesfinanzministerium").