Europe's hotel landscape is quite a bit different from ours: some players are the same, but with very different positions. Knowing the hotel landscape is increasingly important as big chains slowly but surely edge out the funky independent "mom and pop" hotels and bed & breakfasts to which budget travelers formerly gravitated as a matter of course.
For whatever reasons, most of the low-cost chain action is based in France and the United Kingdom and is led by four large chains: French-based Accor (www.accor.com/en.html) and Groupe Louvre (www.louvrehotels.com/en) and the British Premier Inn (www.premierinn.com) and Travelodge (www.travelodge.co.uk). The majority of their locations are in France and the UK, but they're also steadily spreading around Europe.
There is really no U.S. equivalent to the very bottom end of the European budget hotel market:
Accor's Formule1 brand (re-branded as Hotel F1 in France) is iconic: rooms of about 100 square feet, one standard double bed plus mini-bunk for a kid (twin beds in a few locations), washstand and TV. Toilet and shower are across or down the hall. Check-in and room access are completely automated - pay by credit or debit card, then use access codes to get in the hotel and into your room. You can stay there without ever encountering an employee. Rates in small cities start around 30 euros (about $40) per night; breakfast is extra. Formule 1/Hotel F1 is concentrated in France, but operates at least some hotels in Belgium, Germany, the Netherlands, Spain, Sweden, Switzerland, Switzerland and the UK.