Officially he...

-appoints and suspends the chancellor.
But he just appoints the one, that got elected by the parliament and the person, who the parliament elected, is the one of the party with the most popular votes in the parliament.

-appoints and suspends the ministers.
The chancellor proposes them to him, but basically he just makes the chancellor's decision official.

-signs all laws.
He can't change them, he can just say, that he thinks, that they aren't constitutional, but if the constitutional court says, it is, he just has to sign it actually.

-shakes hands with other heads of states and talk to the public on television or in press conferences at special events.

Basically he is, even though officially the "number one" of Germany, just an obsolete organ, that just does, what the other organs tell him to do.
That's also the reason, why he doesn't get elected by the people. If the people would elect him directly, he would have much more power than today most likely.