There are a few different ways to be living permanently in Germany. Depending on your country of origin, you can apply for a settlement permit. Furthermore, after eight (8) years of permanent residency, you can apply for German citizenship through naturalisation and enjoy the same rights as other German and EU citizens.
Residence for EU citizens can be taken up by simply registering in Germany with your passport or national ID card. The European Union freedom of movement rules allows EU citizens to work, study or do training in Germany without a VISA or residence permit.