To "exchange" two things means to switch them:
We exchanged business cards, but that was the last I ever heard from her.
At a store, "exchanging" an item means returning it to the store, and getting something else from them instead.
If it doesn't fit you, let me know and I'll give you the receipt to exchange it.
This is different from "returning" an item, which is when you give the item back to the store and they give you back your money.
In the U.S., most clothing stores will allow you to exchange clothes that you buy for something else. Other stores like electronics stores will also let you exchange something if it's broken. A lot of stores will also allow you to return items, though some places only let customers exchange their items for store credit. This is money that can only be used in that store.