The Minijob (also called a €603 job) is the simplest form of side income: you earn up to €603 per month, and the money lands essentially 1:1 in your account. No wage tax, no social security contributions for you. The €603 threshold is linked to the minimum wage (€13.90 × 10 hours × 4.33 weeks).
Why is the Minijob so attractive? If you already have a main job, you can earn a Minijob tax-free on the side. The €603 is not added together with your main salary. If you have two Minijobs, however, the second one is combined with your main job and taxed normally.
The catch: you build up practically no pension entitlements. By default you are subject to pension insurance (earning about 0.16 earnings points per year – after 40 years, that would be a meager €6 monthly pension), but many opt out. Then even these minimal entitlements disappear. For students, retirees or as a short-term side income, the Minijob is perfect – as a permanent income source, however, you're giving up retirement savings.