Harmonicas are always tuned in a certain key. Does that mean that I can only play songs in that key on it or that it is just harder to play something written in a different key?
Diatonic harmonicas are capable of playing in multiple keys.
This is done a few of different ways. The most common type of blues playing is called "cross harp," which involves using a harmonica in one key to play in a different key (usually a major fifth away). For example, it is common to use a C harmonica when playing blues in the key of G.
You can also use subsets of the notes from the diatonic scale to create pentatonic scales. For example, if you have a C harmonica, just use the notes E,G,A,B,D and you are playing the E minor pentatonic scale. You can also get the A minor and D minor pentatonic scales from that same harmonica.
You can reach other tones by bending the notes. This is a more advanced technique.
There are also chromatic harmonicas, which can play all the notes of the chromatic scale.
Lastly, there is a professional harmonica player named Howard Levy who plays a diatonic harmonica chromatically. He does this by using extremely advanced bending techniques. I've never heard of any other player capable of playing like Howard Levy.
So, to answer your questions: No, harmonicas can play in multiple modes. Yes, you can play songs in other keys, but it will be harder.
If you're talking about the 10-hole 'blues harp', the answer's 'yes and no'! You get the notes of one major scale. Obviously you can play diatonic tunes in that key. But we're in blues country. Use an F harp for a blues in C, that Bb will come in very useful! Or use a C harp for a blues in D, now you've got the flat 3rd and flat 7th. And notes can be bent. And there are other tuning schemes than a plain major scale...
And, of course, there are chromatic harmonicas. Larry Adler rather than - oh take your choice!