A few notes to supplement sauron's writeup on the general principles, from the trader's point of view
VAT and sales taxes using identical principles (GST, etc.) are also applied in many countries outside the European Union, but what follows generally applies to the EU Member States' versions which have fairly uniform principles. Within the EU the minimum level for the normal rate VAT is currently 15%, although it may be adjusted up and down by national governments above that level; there is pressure to increase the concessionary rates; the UK's zero rating for certain essentials is now an exception which cannot be applied elsewhere.
Traders (individuals or companies) are required to register for VAT with the appropriate authorities if their turnover reaches a set figure within a set time, varying from country to country; in most countries this limit is low enough to include all but fairly trivial part-time business activities, and may run in tandem with other business registration requirements but in the UK it is fairly high (£55 000 in any 12 months at the time of noding) and small traders, especially those whose customers are mainly private individuals and who do not spend large amounts on VATable business expenses, may well choose not to register, thus avoiding some paperwork and making it possible to reduce the price to the consumer. If your customers are mainly other businesses (who will be able to reclaim VAT you charge them and thus won't care about the VAT-inclusive price difference), or you are obliged to buy goods for resale or incur other expenses on which you have to pay VAT, it will probably be in your interests to register even if you are under the threshold, although do bear in mind that the reporting requirements mostly come with draconian and non-discretionary penalties; a VAT return is not desperately complicated but you do need the self-discipline to get it in in time, or to pay someone to remember for you...
Traders effectively serve as the collector for the VAT incurred at their stage of the operation. To all intents and purposes this involves reporting (drawing up a monthly or quarterly VAT return and maybe a listing of the details of large transactions or high-turnover clients) the total VAT you spend on business expenses and the amount you charge on sales, and paying over (or, less commonly, being refunded) the difference between the two. National rules vary as to whether this is carried out on a cash or accrual basis (i.e. on money actually received on or invoices issued).
VAT is not charged (or can be refunded) on goods or services exported from the European Union. For crossborder sales to other EU countries, the VAT in force at the point of sale or the place of supply of the service (which can be complex to determine for work carried out remotely over the Internet...) is charged, except where the customer is also VAT registered; in that case the "chain" is broken at the border - this avoids the sort of accounting gymnastics that would be required to reclaim or deduct VAT from different countries' systems. However, if you do a very large amount of business in another EU Member State, you may be required to account separately for VAT there in additional to your local requirements.
SMALL PRINT: If you are considering going into business at some level, you're almost certainly best off finding a qualified accountant to handle this (and other) stuff for your particular circumstances; I am only in a position to be offer the faintest approximation of authoritative advice for translators contemplating operating on a freelance basis under the laws of Belgium (and that advice is mostly "Don't.") Now, where are those forms? What is this "special account"? Why have I been fined €500? What do I do about this credit note? Why do I have to pay the VAT up front that my customers pay at 60 days end of month? Aaaaaargh.