Up to my ears in stuff right now, or I'd try to give a comprehensive answer to Pip's questions. But here's at least one:

"Resolving" refers to relieving a "dissonance" in the harmony. Dissonance is when two notes seem to clash in an unpleasant sound. Say C and Db played together. Like fingernails on a blackboard. But—sometimes you want a touch of dissonance to create tension in the harmony, to indicate "I'm not done yet! Stay tuned!" A sort of musical "cliff-hanger."

An everyday example of this: say you're playing in the key of C. One of the chords commonly used in that key is a G7. Now a G chord is "consonant," which is to say, the three notes that make up the basic chord, G, B, and D, sound good when they're played together. But if you add an F on top of the other three notes (seven scale steps above the root—G—hence, "7th"), the F and the B in particular don't quite get along. They form a "diminished fifth," which is a dissonant interval that calls for "resolution." When you change from a G7 to a C, the B in the G7 moves up a half-step to a C, and the F moves down a half-step to an E, "resolving" the dissonance.

The G7 chord is the "dominant 7th chord" in the key of C, and it provides the "drop the other shoe" effect at the ends of verses and the end of a song.

I hope I haven't totally muddied the issue. This stuff can be very confusing unless one goes at it systematically, beginning with the very basics (what is a scale, what kinds of scales are there, what is an interval, what is a chord), moving bit-by-bit into the messier stuff.

But it's well worth the study!!

Don Firth