Oil-based paint is more durable, but it takes longer to dry, and cleanup requires turpentine or paint thinner (White spirits). Oil-based paints are made with either alkyd (synthetic) or linseed (natural) oils. Alkyd paint is more common because it is less expensive and tougher. Oil-based paint is good for trim work because trim takes more abuse over time than do walls.
Water-based paint is easier to work with and dries more quickly, but it isn't quite as durable as oil-based paint. Water-based is good for general painting projects such as walls and ceilings.
Paint comes in a number of different finishes, from flat to high gloss. Some paints also contain enamel, an additive that makes the dried surface harder and less porous.
If you're working on unfinished walls, be sure to apply a coat of primer before you begin painting. Use water-based primer if you plan to use a water-based paint. Oil-based and Oxide primer works well with oil-based paints.