A basis swap is a type of swap in which two parties exchange the interest payments based on two floating rates.
The basis swaps are used to hedge the interest rate risk arising from the borrowing and lending at two different floating rates.
For example a bank might be lending at an interest rate tied to Libor but itself borrows at Constant Maturity Treasury rate. The hedge the interest rate risk, it may enter into a basis swap to pay at LIBOR rate and receive at Constant Maturity Treasury rate.
The pricing of basis swaps is similar to currency swaps, except that there is no exchange rate involved.
There are also cross-currency basis swaps, in which two streams of money market floating rates of two different currencies are exchanged. The cross-currency basis swaps may or may not involve exchange of notional at the beginning and end of the swap.