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. Receipt can also be used as a verb, as in, "Are you in receipt of the package I sent you last week?" That's a fancy way of saying, "Did you get the package I sent?"