§ 32a EStG describes how the income tax was calculated in 2009. In 2018 an income less than 9,000 €/year is tax-free. Incomes up to 54,949 € are taxed with a rate progressively increasing from 14% to 42% or better from 14.77% to 44.31% because of the solidarity surcharge which has to be paid. Over 54,950 € the tax calculation is simple: take 42% of the income minus 8,621.75 € and then add 5,5% (of the calculated tax) as solidarity surcharge. In 2018 above 260,532 €/year the (2007 new created) income tax ("riche tax") is 45% (tax above 260,532 €: 45% of the income minus 16,437.7 €).
There are still many questions coming along about the German progressive tax tariff. A program plotting the German income tax burden rate against the income could be found under Diagram of the German income tax burden. This Diagram shows, that the basic tax free alowence (9,000 € 2018) is part of the tax tariff and therefore already taken into account to calculate the income tax.
For married couples the same formula is used but with the difference that for only the half of the income the tax is calculated and afterwards this tax is then doubled, which is an advantage in the progressive tax rate region (2018: from 18,000 - 109,898 €).
The German tax law allows to deduct several expenses from the income before taxation. For every child (under 25 and still in professional training, living in Germany/EU) a lump sum of 7,428 € (2018) is deductible from the income, but this influences only the social surcharge (and church tax) and not the income tax. This is because of the child subsidy of 2,328 € (2018) every single-parent or married couple receives having a child (under 25 - still in professional training, living in Germany/EU) and which is normally more favourable then the lump sum 7,428 €. But for parents with a total income above 68,941 € (2018) this is different because then the tax gain for 7,428 € on the couple tax tariff is more then the child subsidy of 2,328 €. In this case the tax authorities will change this to the taxpayers favour. The program shows when this is the case.
Before starting a business it is recommended to inform the tax authorities. The official form to do this could be found under Fragebogen zur steuerlichen Erfassung at the pages of the Bundesfinanzverwaltung (only in German).
Non-residents (not living in Germany) with income in Germany (such as property let) are subject to limited income tax liability. They could only deduct expenses which relates to the income (such as renovating the appartement) and have not the basic tax allowence (tax free part of the income) of 9,000 € (2018). Because the basic tax allowence is integrated in the income tax tariff one has to add first this amount to the income before calculating the income tax (§50 EStG)