Asset Swap

Go to a garage sale and instead of paying $2 cash for that juicer, you negotiate a trade for the toaster oven that's been driving around in the trunk of your car for the past two months that you never quite found time to drop off at the Salvation Army. That's one form of asset swap.
There's a more complicated Wall Street version as well. A typical swap involves trading cash flows (See: Agreement Value Method). They are executed through a derivative contract and involve trading, say, the payments on a fixed-rate investment for the revenue from a floating-rate investment.
In an asset swap, instead of trading cash flows, an actual asset changes hands as well.
For example, a buyer acquires a fixed-rate bond (that's the asset being swapped). Then, instead of getting the regular interest payments from the bond, the buyer then swaps these fixed-rate payments for a floating-rate payment based on prevailing interest rates.
There are many reasons why an asset swap might make sense, but the main goal is to hold a long-term asset while having the payments match the fluctuations of interest rates. An example typically given is a bank looking to have revenue coincide with short-term liabilities of their depositor accounts.