VAT can be loosely compared to the US sales tax. It is added to every purchase, service, and more or less everything that requires payment. But since you'll probably be only shopping - consider it as "something like a sales tax". In Europe, prices are always quoted with the VAT already included.
The VAT itself can be up to 25% in some places, so its good to know that you can get some of it back, when you leave with your purchase.
For some (larger) purchases, you get the VAT back at the airport. It has to be over a certain amount per each receipt, and the store has to fill out a special VAT return form for you.
Make sure, if you're paying more than say 50-100 EUR in a single transaction, to ask the seller for that form. Sellers that provide this form have stickers showing "Tax Free Shopping" or "Vat Refund for Tourists".
Here's an example of how to ask for the refund:
Everywhere else its similar.