Most shows that are not Mickey Moused have a stock set of instrumental BackGround Music (BGM) to use as themes for particular characters or sequences, or simply to set a mood. These usually fill up the bulk of a show's soundtrack, aside from Theme Songs and (in the case of anime) Image Songs. Music in Video Games can also be referred to as BGM, as some (often older) games have music playing in the background at all times, rather than a blend of music and ambient sound effects.
The pieces that are most closely tied to characters or events are Leitmotifs.
When it's not so much in the background, see Left the Background Music On.