No qualified vocational training
If you have
- a specific job offer and
- the general requirements to obtain a residence title are fulfilled (passport, the costs of living are covered during the stay, no grounds for expulsion exist),
a residence permit to engage in employment can be granted (residence title according to § 18 AufenthG).
Employment with or without qualified vocational training generally requires the appoval of the Federal Employment Agency.
There are also approval-free employments, which are according to German Employment Regulations among others activities of executive managers and professionals in science, research and development (§§ 3, 5 BeschV, German Employment Regulations).
For a stay for the purpose of gainful employment according to § 18 AufenthG times of employment can completely count for the settlement permit according to § 9 AufenthG. Also half of the duration of academic studies and vocational training in Germany is taken into account, see § 9 IV No 3 AufenthG.