It depends on your exact itenerary, how how long you're staying, what you plan to do, etc. Just remember that things get more expensive the farther north you go--budget more for a trip around scandinavia than a similar trip around Italy and Greece. Here are some useful resources for planning a trip that you can also use to work out about how much your trip will cost:
Getting to Europe:
www.statravel.com - if you're a student or under 26 this site can get you excellent deals on airfare
yahoo travel (or experida, priceline, etc) also may have good deals
also check out the websites of the individual airlines, as some of them have good deals as well that don't wind up on discount sites.
Just jumping on Yahoo travel it looks like you could get from Washington DC to London and back for $370.
Getting around IN Europe:
check out Interrail/Eurrail (just google it) in order to get your classic rail passes. Note that rail passes aren't always the most cost effective way of getting around--especially if you're covering a small area or only seeing a couple of places. A ticket good for 10 travel days travel in 3 weeks cost 359 Euro when I did it a year ago (so now 400-500 US)
Also check out ryanair and easyjet-- just remember that you will have to spend plenty of money to take luggage, and getting to and from out of the way airports can be expensive and time consuming.
especially if you're going by yourself, stay in youth hostels, save money, meet interesting people. These websites allow you to book a youth hostel bed just about anywhere. Their user generated ratings are also pretty accurate.
Stuff to do:
Get a lonely planet guide (they're ridiculousy useful) and also will let you know how much it costs to get into the Acropolis, the Colosseum, the Louvre, a train to the Grindelwald, etc.
And of course budget a bit extra for food, souveniers, something going wrong, etc.
I spent about $3,000 US over the course of three weeks backpacking through France, Italy, Greece, Switzerland, Germany, and the Low countries, though this was when the dollar was weaker than it is now.
You can probably figure abou $100 US/day plus or minus, depending on so many factors....plus your airfare and ground travel.