Most shows that are not
Mickey Moused have a stock set of instrumental
Back
Ground
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.
The pieces that are most closely tied to characters or events are
Leitmotifs.
For when it's not so much in the background, see
Left the Background Music On.
See also
BGM Override,
Variable Mix,
Diegetic Switch.
Specific instruments and
Standard Snippets are used to create a
Mood Motifs.
For further exploration of this topic, listen to Anna Russell's "The Ring of the
Nibelungs (An Analysis)".