Stainless steel yields a brighter sound and resists corrosion better than Nickel, in most cases are usually rougher on hands and frets. Nickel is usually smoother, less top end which gives the illusion of more midrange, and easier on hands and frets.
Round core = more flexible, will require more relief in the neck or higher action to prevent buzzing. Sounds better in my opinion though and I still have my action way down load.
Hex core = stiffer, can get away with less relief in neck and lower action.
Roundwound = outer wrap is of a round shape. Brighter sound, more sustain usually.
Flatwound = outer wrap is flat. More mellow tone and less sustain.
Tapewound = outer wrap is nylon on flat or roundwound. Less tension, cool sounds that can be hard to describe.
Pressurewound = outer wrap is compressed to an oval shape before wrapping on string. Between rounds and flats, closer to rounds in tone and feel.
Groundwound / Halfwound = outer wrap is ground flat. Between rounds and flats, closer to flats in tone and feel.
Taperwound = thickest strings (usually B, sometimes E and A as well) have a slim section that passes over the bridge before coming to their full thickness.