Fulls: Simply put, this is just any flip type, with a twist. Within tricking, it usually refers to a backflip with a full twist (back full), or what can be argued to be more of an inside flip with a full twist, such as is commonly executed from a cartwheel (cart full). In either case, the goal of the performer is usually to reach a high degree of inversion while twisting.
Corks: By far the most popular family of tricks in tricking, corks are simply back fulls from a gainer setup that utilizes the inside kicking leg to swing. Another way to look at it may be that they are back fulls from one leg. Corks are most often initiated from a backside stance, and are swung into from a complete landing. Some trickers differentiate fully inverted gainer-fulls and a more flat-spun cork, but regardless of the level of inversion, the general technique and family of tricks is the same.