Staying in a student hostel is a great, cheap way to travel. You're not going to get 4-star hotel accommodations, but generally student hostels are clean, safe, and inexpensive. Plus you can meet really interesting people your age from all around the world.
Sleeping quarters are usually dorm-style and may not be segregated by gender. Many hostels include breakfast, hot showers, and wi-fi access. There may be a kitchen so you can make your own food, and laundry facilities.
There's a great network of student hostels in Europe. In general, you'll pay less than $50 a night to stay in a hostel, about a quarter of what a moderate hotel charges.
For a list of hostels in Europe along with great tips on getting around and saving money, see Backpacking Europe: Hostels and Travel Guide on Amazon.