Someone please explain these things to me and how you would figure out how to apply modulation to playing or songwriting. I don't understand these at all and the threads here haven't really cleared anything up.
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that's more than anyone can explain in one sitting over the internet.

The very slim basics are that passing tones/chords are notes that are technically out of key/scale, but are used in transition or for color.
Resolving chords are are a series of chords that brings an arrangement back from a tension or other passage back to the key center/tonic chord.
Blues layed down some decent definitions for those two terms; I'll let you in on modulation.

Modulation is basically a fancy term for the process of shifting a song into a differant key. There are numerous ways of doing this; you can play a non-scale tone for a number of bars to familiarize the listener with the pitch, which may later be the root of your next key, or significant pitch in that new key. There a ton of other ways...just wikipedia for a definition of all differant types of modulation.