rockeyracoon covered it for the most part.
You can view your checking account's transactions via your bank's website (don't know how most banks work but mine had me create my login information when I opened my account) but only be granted the name of the website the purchase was made at and the amount spent.
For a record of the entire purchase the source website (ex. Amazon) will usually send a confirmation email right after the order is completed which includes the exact item purchased and return/complaint information, basically a virtual receipt.