An agreement to, for example, pay fixed 5 year, and receive floating 3 month Euribor. A means by which banks would for example hedge fixed rate customer loans. The bank receives fixed 5 year from customer, pays fixed 5 year on the swap, and receives floating 3 month. Net position is a 3 month floating rate received.
There are absolutely infinite variations to this very basic instrument, where the basis (floating side) can change, or can have floating against floating, or pay/receive in different currencies.
It's where two companies decide they can get a better deal by swapping contracts, company a has a fixed mortgage and company b has a variable mortgage. Because of credit checks etc say the co's can make better savings by swapping with each other instead of striking their own deal with the bank.
Because the biggest risk is default from the other party, nearly all swaps are done through a bank.