This is called a turn. The 'basic' version would be written without the accidentals, and the player would play the first note, then quickly play one tone (note) above, the main note again, a tone below, the main note, and the resolve one the final note. The accidentals clarify exactly which notes to "twiddle" to.
The turn can be either directly over a note, or between two notes. (It looks to me like this is intended to be between two notes.) Over a note, the whole value of the note would be used for the twiddle, between two notes, the 'twiddle' would be squeezed into the gap!
I'd play this example like this:
This is a trill. Quickly alternate between the marked note and a tone above, for the entire duration of the note.
An Acciaccatura (a.k.a. crushed note, grace note.) Play the small note as quickly as possible, just before the next main note, as a little lift/fall to the main note on the beat.
This notation I'm less familiar with, and not aware of it's specific name. (NReilingh provides a much better explanation in the comments.) Here, the small notes are to be played 'more freely' over the sustained chords underneath - not strictly in time. This youtube recording is a good example. (This part is around 3.40)