Generally speaking, fats and oils, at least in the kitchen, are triglycerides. You can find a brief description here:
http://en.wikipedia.org/wiki/TriglyceridesIn water, with a small amount of either acid or base, the most general reaction is hydrolysis of the esters. This generates long chain fatty acids and glycerol.
As Discodermolide pointed out, there are also additional reactions that may occur based on any functional groups present on those fatty acids.