In concrete terms, the president has a very limited role. He or she (until now, always “he”) has some powers and a role during the change of government but even those are mostly formal.
He does not really have the power to dissolve the Bundestag (the lower chamber of parliament) when he sees fit, he can only do that if the parliament fails to agree on a chancellor or after a failed vote of confidence. (Incidentally, there are some legal debates on what exactly the role of the president is in those case especially when the failed vote of confidence was actually engineered by the chancellor and his or her majority to provoke new elections.)
The president also signs the laws before they are enacted and could in theory exercise a kind of “veto” that would need to be resolved in one way or another. But he can only do it under limited conditions (see comments) and seldom uses that to exert influence in practice. It does however happen (mostly the president indicates his concern and the law does not make it to his desk). Presidents have also referred statutes to the federal constitutional court.
Beyond the formal competences defined in the constitution, the president is expected to be politically neutral. The current president's outspoken views is a strong break from tradition. Even the fact that he merely voiced an opinion about a coalition in one of the federated states has been criticized.
I am therefore not sure how the idea of “separation of powers” applies here. Both the chancellor and president can be regarded as part of the executive branch but the chancellor and the government effectively have almost all executive powers and an important role in preparing legislation too.
If you must draw an analogy, the chancellor could (with lots of caveats) be considered as the counterpart of US or French presidents (when they have a parliamentary majority, which has not always been the case in those countries) and of course like the prime minister in constitutional monarchies (the UK, Spain, Belgium, the Netherlands, Norway, Denmark, Sweden, etc.). Like in other parliamentary republics (Ireland, Portugal, Italy), the president does have some modest role but the title should not mislead you in thinking it is in any way similar to the president of the US.