There's no getting around it: Europe is expensive. Compared to other popular backpacking destinations, prices in most European countries are many times higher, making budget travel difficult. However, it is possible to see Europe without breaking the bank.
Plan ahead. Research the countries you plan to visit before you leave. Get an idea of what current prices might be, and check out the various methods of travel between your destinations. The more you know before you leave, the more prepared you'll be when you get there.
Bring a guidebook. The Lonely Planet, Rough Guide, and Let's Go series are all targeted towards budget travelers. They'll identify cheap hostels, budget food options, and other money-saving tricks. You might not follow all of the information, but it will give you a place to start when backpacking in Europe. After all, if you arrive in Florence knowing that the regional trains are cheaper than the Eurostar trains, you'll save money immediately.
Sleep on the cheap. Hostels in Europe are downright expensive compared to hostels in Southeast Asia and Central America, but they are considerably cheaper than European hotel rates. If you are traveling in peak season, it is wise to make reservations before you arrive, as the cheapest places fill up fast. Look online at sites like HostelBookers, HostelWorld, and STA Travel to get an idea of prices and to book rooms. If you're open to it, some cities offer campground facilities for even cheaper rates. Programs like the CouchSurfing Project and the Hospitality Club consist of a network of members who are offering free accomodation all over the world; if you're brave enough, sign up and sleep on a stranger's couch for a night or two.
Stay in for dinner. Dinners in European restaurants are often two or three times more expensive than lunch prices. Go out for lunch, when prices are cheaper, and buy street food or fresh market vegetables, bread, and cheese for dinner. To keep food costs low, shop like the locals: seasonally, and often. And think about it: what's a more European experience than a picnic of wine, bread and cheese in a park with a view of the Eiffel tower?
Use low-cost airlines. Once upon a time, Eurail passes were the cheapest way to travel around Europe. With the introduction of low-cost airlines like RyanAir and EasyJet, and WizzAir, this is no longer always the case. Budget airlines offer fares as low as one Euro (watch out for taxes and fees that drive the price up) to hundreds of destinations in Western and Eastern Europe. Research prices for your itinerary: it might be more cost-effective to use regional train travel for short distances and air travel for longer distances.
Go in the offseason. The summer months are the prime European backpacking season, and cities and major attractions will be overrun with tourists. If you can, go in the offseason, when prices are lower, lines are shorter, and locals are more accommodating. You won't have to fight for hostel rooms, and you'll be able to experience more of the local culture.
Find discounts. If you are a student, youth under 26, or a senior, you can get a discount card for hostels and travel in Europe. Organizations like STA Travel offer discounted travel options, and some hosteling organizations, like Hosteling International, offer discount cards with membership. The money you save at each place may seem small, but it adds up over weeks and months.