You know that slim, white piece of paper the grocery store clerk gives you that lists everything you bought, how much you paid, and the change you received? That's a receipt.
A receipt is proof you bought and paid for something. At most stores, you need it if you're going to return what you bought. People collect receipts on business trips, so their employers will pay them back for what they purchased while away. Less commonly, receipt can be used as a verb, as in, "The clerk receipted my purchases," meaning that the clerk gave written proof that the items were paid for.