If your mobile operator supports it then purchasing apps on Windows Phone 7 is very straightforward (if your operator does not support this then you will need to register a credit card via the Zune software on your desktop before you can make any purchases).
After purchase the app will automatically download and install itself onto your phone (you can see progress by opening the Marketplace App on your phone and selecting the ‘downloads’ link from the bottom of the page.
Finally, you will receive a receipt for your purchase – either as a part of your network operators bill, or via email for credit card purchases
Below is an example of the receipt I received when I purchased SNAP!: