VAT = Value Added Tax where the tax is incorporated into the price of the item. VAT applies to goods AND services. It is called different things in different countries (TVA, MwSt, IVA etc.), but they all translate as Value Added Tax.
VAT is a SALES TAX, but in all European countries I know of the advertised price (marked price) must include all taxes. Otherwise it is fraudulent advertising.
DUTY FREE = Products purchased in duty-free shops which have no taxes paid. These goods must be exported intact. Duty free shops are usually in Airports. Whether they are taxed in the country of arrival depends on whether you are importing more than a certain value (which varies by country). The import taxes applied by the country of entry are the same whether the goods were bought duty-free or duty-paid.
There is no Duty Free when travelling between EU countries, because the EU is a Customs Union. But on the other hand there is no Duty or tax to pay on goods purchased tax paid in another EU country, so long as they are for personal use (will not be sold).
VAT REFUNDS = What the link above says is, in my opinion not 100% correct.
VAT is not charged on exports outside the EU. If VAT has been paid (the goods have been purchased in a normal shop, not a duty-free), it is possible to re-claim the VAT, but it is not easy. You can only claim VAT back on goods which are exported intact, not on hotel or restaurant bills.
The VAT must be reclaimed by the shop that paid it, they then reimburse you, usually less a handling charge. Because you don't want to go back to the shop to get paid, you must pay for the goods with a credit card, and you are reimbursed later with a payment to the credit card. Only some shops will do the paperwork, and only for big purchases. It is only worth the hassle for the shop if they have enough non-EU tourists making big purchases. if not, hard luck.
- When you buy the item, tell the shop you want to claim back the VAT. They provide you with an export form.
- When you leave the EU (airport or land border) you show the goods and the form to customs, and they stamp the form to certify that the goods have left the EU.
- You then send the form back to the shop. They re-claim the VAT and credit your card.
- You must get the form stamped when you leave the EU. For example, if you buy something in London, then travel to Paris, then from Paris to the USA, the form needs to get stamped at the Paris airport. But, if you travel London - Paris - Switzerland and fly back from Zürich; Switzerland is not in the EU, so you leave the EU at the French-Swiss border. But it is not very easy to stop at a French customs post if you are on a high speed train! :-)
- The goods must be exported INTACT AND UNUSED. If flying, you must have them in your hand luggage so the customs person can inspect them and certify they have been exported. Often they don't bother, but you may be unlucky.
- This is only possible for non-EU residents, so you must show your passport or other proof that you live outside the EU to the customs person.