Granting asylum to political refugees and protecting refugees according to the Geneva Convention is and remains an obligation and the responsibility of the state written into the German constitution and demanded by international law applicable in Germany. However, due to very restrictive reforms of law and procedure it is in fact very difficult to be granted asylum and refugee status in Germany. Furthermore, the European refugee-blocking system that is being implemented in the EU and its border states makes it increasingly difficult to even reach Germany as a refugee.
The defining phase of an asylum application is right at the beginning, most importantly the first interview with the Federal Immigration and Refugee Office (BAMF). It is therefore highly advisable to look for assistance early on, be it a lawyer or an experienced NGO like Amnesty International or others.
As an asylum seeker, you will get a Permission to remain (Aufenthaltsgestattung), that will be renewed for 6 months, as long as the asylum proceedings last.
If your application is rejected and if the decision is final because all appeals habe been lost, your right to remain in Germany as an asylum seeker is lost as well. You will be asked to hand back the Aufenthaltsgestattung and leave the country within a month. However, if you cannot leave and if the authorities cannot deport you for legal and/or practical reasons, you will get a Stay of deportation (Duldung)
When a Duldung will not be given or renewed, you might receive a Grenzübertrittsbescheinigung with a duration of a few days only that basically just shows that you are registered with the authorities and allows you to leave Germany without being arrested at the border.