A swap is an over-the-counter agreement between institutions to "swap" one thing for another, usually the cash flow related to interest-bearing instruments.
Given the negotiable and over-the-counter nature of swaps, there are many permutations and manifestations of this concept. The most common is the interest rate swap, in which the counter-parties agree to pay the interest due on principal amounts which are not exchanged.
These are done as a hedging strategy against interest rate risk, and are transacted intercontinentally, usually, between one party who has a fixed interest rate and another party which has a variable interest rate. In such a way, institutions can diversify their exposure to interest rates around the world.
Other types of swaps include Equity Swaps, Currency Swaps, Commodity Swaps, and Credit Swaps.