Let's assume it's a Blues harp, with probably 10 holes, rather than a full blown(!) chromatic harmonica. It's marked as a C harp, as all the notes played when blown or drawn will come from the C diatonic scale. Unless they're bent - more later.
To play along with another instrument, it could be played straight, as in both are using key C for the song. Or, it could be used as a cross harp, where the harp is in key C, and the other instrument plays in key F. That way, there are several draw notes which can be 'bent' to produce Blue notes (in key C). It's easier to bend draw than blow notes on a harp.
So, in answer to the actual question, you'll need several harps in different keys. It's what harp players do - they'll use a cross harp for a given key P4 up from the harp they need. The good ones will actually, say, for a song in key A, use two harps - a D harp for the Bluesy bits,(cross harp), and an A harp for the straight parts. Easy way to work out - whatever the main chord is when you draw on the harp, that's the key of the song it'll work best in.
There are also harmonicas available in minor keys, but maybe we don't need to tread that far just yet.
As far as playing chords a C harp will play a good C major triad, a nice G9 triad, and with very careful tongue placement, other 'chords' (dyads) but only those which use notes diatonic to their own key.
Another option is the aforementioned chromatic harmonica - the chromonica - with a button on the r.h. side. But this will, for a marked C chromonica, play in C/G, or C♯/G♯. Probably not favourite alternative keys, though! Having said that, a good player will use a chromonica to solo in all available keys - Toots Thielmans and Larry Adler both spring to mind.