A basis swap is an interest rate swap where the floating leg references a different index than the fixed leg. The most common type of basis swap is the USD LIBOR/OIS (London Interbank Offered Rate/Overnight Index Swap) basis swap. In this type of swap, one party pays a fixed rate in exchange for receiving a floating rate based on LIBOR and the other party pays a floating rate based on OIS.
The rationale behind why someone would want to do this kind of swap is that there are different expectations for future interest rates among market participants. For example, banks tend to have a more positive view of future interest rates than non-banks because they can take advantage of higher rates by lending out money at a higher rate. As a result, banks will tend to pay a higher fixed rate in a basis swap than non-banks.
Basis swaps can be used to hedge interest rate risk or to speculate on future interest rate movements. For example, if an investor believes that LIBOR is going to increase relative to OIS, they could enter into a long USD LIBOR/OIS basis swap. This would mean that they would pay a fixed rate and receive a floating rate based on LIBOR. If their expectations are correct and LIBOR increases relative to OIS, they will profit from the trade.
Basis swaps can be traded over-the-counter or on exchanges. The most common exchange-traded basis swaps are based on LIBOR and EURIBOR (Euro Interbank Offered Rate).