Resolution in western tonal music theory is the "need" for a sounded note and/or chord to move from a dissonance or unstable sound to a more final or stable sounding one, a consonance. Resolution has a strong basis in tonal music, since atonal music generally contains a more constant level of dissonance and lacks a tonal center to resolve to.
An example of a single dissonant note which requires resolution would be, for instance, an F during a C major chord, CEG, which creates a dissonance with both E and G and may resolve to either, though more usually to E (the closer pitch). In reference to chords and progressionss for example a phrase ending with the following cadence IV-V, an imperfect cadence, does not have a high degree of resolution. However, if this cadence where changed to (IV-)V-I, a perfect cadence, it would resolve much more strongly by ending on the tonic I chord.
Dissonance, resolution, and suspense, can be used to create musical interest. Where a melody or chordal pattern is expected to resolve to a certain note or chord, a different but similarly suitable note can be resolved to instead, creating an interesting and unexpected sound. For example, the deceptive cadence.
The concept of "resolution", and the degree to which resolution is "expected", is contextual as to culture and historical period. In a classical piece of the Baroque period, for example, an added sixth chord (made up of the notes C, E, G and A, for example) has a very strong need to resolve, while in a more modern work, that need is less strong - in the context of a pop or jazz piece, such a chord could comfortably end a piece and have no particular need to resolve, see: fadeout.