One way of saving money is to never travel alone. Two people sharing a hotel room almost cuts the expenses in half. If you don't mind the dorms from the youth hostels where you share one big room with other 6 or 12 fellow backpackers you've never seen before and probably you will never see again, then you have my blessing and my assurance that this is the cheapest accommodation you will ever find in Europe, unless maybe if you stay at a friends house. These dorms can be mix or unisex, depending from one hostel to another. Dorms might be suitable for single people, groups of friends and people that don't have issues about sleeping in the same room with strangers. Unfortunately for my bank account, I do have issues regarding the dorms and never spend the night in one. And that's still 500euro a week!
By the time I got to backpack Europe I was already married and traveling with my husband and staying in a hotel or a private room in a hostel seemed the best choice. It is true that we always spent a bit more on our accommodation than other backpackers, but if you do the math, the safety and privacy you get in return for the small extra we paid, was always worth it. You can expect to pay between 10 and 20euro per night for a bed in a dorm in most European countries. That's between 20 and 40euro per night for 2 people. We usually spend between 42 and 55euro per night for a private room. London and Copenhagen are the only two exceptions that were way over our planned budget. Everywhere else, we happily managed to stay within our budget.
We never booked accommodation in advance, except once or twice when we booked through Etap Hotel and for our last trip to Milan, Lake Como and Turin, which by the way cost us only 350euro/6 days/person, due in part to a fantastic deal we got with RyanAir. We always move boldly from one city to another and though most of the time we travel during the high season, we never had problems finding accommodation.
Most cities have a tourist information office either inside or outside the train station and they are more then happy to book right there and then a room for you. Sometimes you have to pay up to 5euro extra for the service and sometimes they just point you out on a free tourist map where to go for cheap accommodation as they don't work with small, family run Bed&Breakfasts. As long as you keep it more or less on the beaten path, cheap accommodation options are not hard to find in Europe. Once you get off the beaten road, tourist information offices might be difficult to find, but you will never run out of budget options to spend your night.
More articles on the subject of backpacking Europe to come. Cheap food and cheap transportation to be coverd too.