Hostels:

One of the cheapest and most fun ways to stay in Europe is in youth hostels.

Hostels can vary greatly, be sure to search on-line or consult your guide book to find a hostel that matches your needs. Be sure to balance cost, amenities, cleanliness etc. when deciding on a hostel.

Hostel Pro's and Con's

Let's look at some Pro's and Con's of staying in a Hostel.

Hostel Pro's

Hostel Con's

Pretty cheap Sometimes you get what you pay for, hostels are not always the nicest
Generally helpful and knowledgeable staff, many are past-backpackers Lack of Privacy, unless you get a private room which is more expensive
Easy to meet other travelers Shared living means it can get a bit noisy, making it hard to sleep
Many have a cheap bar attached May have to provide your own sheets, or rent sheets for extra money
Access to Internet, Kitchen, Laundry at many  
Many have great locations

I always pre-booked my hostel rooms a couple days in advanced so I didn't have to worry about not having a bed. I would especially pre-book weekends, as these fill up faster, especially in places like Amsterdam, Paris or London.

If you've taken a spur of the moment trip and just arrived in a new city, see if you your guide book has any listings for that city. You can also ask at the information booth at the main train station, most have a list of hostels.

Finding a Hostel:

HostelWorld

I have used Hostelworld.com almost exclusively to book my hostel rooms and I have never been anything but impressed by their service.

Hotels:

Hotels don't have to be prohibitively expensive, especially if you're traveling in a group of 2~4. If you've spent week after week in hostels, think about treating yourself to a night in a hotel, so you can enjoy a really good night sleep and some privacy for a change.

Hotels Pro's and Con's

Let's look at some Pro's and Con's of staying in a Hotel.

Hotel Pro's

Hotel Con's

Hotels are usually nicer You'll also pay more than a hostel room. More affordable if split amongst a group
Privacy, and private bathrooms Harder to meet other people, other guests are usually older
Sometimes Breakfast included Cheaper ones are usually further away from the city center

Finding Cheap Hotels:

Hotels.com

Hotels.com offers thousands of discounted hotel rooms across the world. I used them to stay in Potsdam, Germany for less than the cost of a Berlin hostel.

Octopus Travel

OctopusTravel offers a number of discounted hotel rooms across Europe.

Priceline

Priceline is another major hotel search engine. You might try their bidding system or at least search to see if you can find any cheap hotels.

Other Lodging Resources

There are other ways to sleep as well, some are even free.

CouchSurfing -- CouchSurfing is worldwide network connecting travelers with local people that have a spare couch. It's a great way to both cut down on sleeping costs as well as meet some amazing people. I would highly recommend trying this at least once on your trip.

The Budget Traveller's Guide to Sleeping in Airports -- If you really need to, you can try being a vagrant for a night and sleep in an airport. It might not be comfortable, but hey, at least it's free.