The exchange of two floating rate interest streams of different payment basis. The different basis, can be either the payment frequency, day count or business day convention. As both streams are floating rate and therefore are relatively insensitive to changes in market interest rates , the streams have low NPV (net present values) and so basis swaps typically have minimal valuations (although different payment frequencies can lead to accrued interest balances).
Cross Currency Basis Swaps involve exchanging floating rate streams of different currencies These types of basis swaps are far more common as there is an intrinsic value to the swaps. The swaps can either have an exchange of notional on maturity or no exchange of notional, exchanging the notional on the maturity of the swap is the most common form as the swap is used to hedge an offsetting loan/investment which will be repaid on the maturity of the swap.