All content copyright © 2010-2018 Frank Revelo, United States copyright office registration number TX-7931345
Unlike the United States, where trails tend to stick to the wilderness, trails in Europe tend to start and end in towns, and to pass through or near towns at least daily in the lowlands and at least every few days in the mountains. It is thus quite possible, and indeed common, for people to walk for months at a time across Europe and never camp once, but instead sleep in hotels or refuges every night.
On the other hand, since hotels can be full or closed unexpectedly, it is always best to be prepared to camp. At a minimum, I would recommend carrying a lightweight sleeping bag and mattress, which will allow you to sleep in refuges or on the floor under a roof (or under the stars in dry summer weather), in case no hotel room is available. I carry full camping gear myself and tend to alternate between wild camping, camping in organized campgrounds where I can wash myself and my clothes, and staying in hotels in the larger cities. I avoid refuges, hostels and other shared dormitory situations, partly due to one too many experiences with snorers and partly because I simply like privacy at night. Organized campgrounds are often crowded and noisy, especially in the summer, so I prefer wild-camping, which is sometimes called "bivouacing". Technically, wild-camping is illegal in most of lowland Europe, but as long as you follow stealth-camping and leave-no-trace principles, there probably won't be any problems. Pitch your tent in the evening in a secluded piece of brush-covered waste land or communal forest. NEVER MAKE A FIRE! Strike your tent in the morning. Don't leave any trash behind. Avoid wild camping when there is a campground or inexpensive hotel nearby. Sometimes kids riding their bikes or villagers walking their dogs in the evening find me setting up camp, but they usually just smile, and exchange a few polite words with me about what I'm doing, then leave me in peace. Only once have the police come by. After checking my passport number via telephone against the national crime database, they informed me that camping wasn't permitted and then left me alone after I said I would be moving on early the next day.
Finding laundromats can be difficult in Europe, and next to impossible in the rural areas I prefer. Therefore I strongly recommend clothing which can be easily washed using cold water only in a sink or shower, especially inner clothing (outer jackets don't normally need frequent washing). Ideally, pants, shirts and underwear would be nylon and socks either nylon or wool/nylon blend. As noted on the clothing theory page, nylon and wool are both easy to wash in cold water, and nylon is additionally fast-drying. Wool/nylon socks are not fast-drying but this is not a problem. Just put them on wet in the morning and walk them dry.
European Ramblers Association shows overview maps of the various long-distance European hiking trails, and also has links to the various national hiking organizations in Europe.
As of 2014, Americans are allowed to spend 90 days in every 180 days in continental Europe (Schengen zone) as part of the visa waiver program, plus an additional six months in Britain. Many eastern European countries are not part of Schengen currently, including Bulgaria, Romania, Ukraine, etc, so visiting those countries is another way to spend more time in Europe. Between 2000 and 2010, France and Spain were not enforcing the 90 day limit for Americans, at least based on my experience and internet research. As of 2007, Greece was stringently enforcing the Schengen rules, with fines of 1000€ (or a 5+ year prohibition on returning to the Schengen zone) for overstaying the 90 day limit by even a single day. Since 2010, I have not stayed in the Schengen zone for more than 90 days per year, so I don't know if things have changed since then. There has been some discussion by the EU of changing the rules to allow tourists to spend up to a year in the Schengen zone, provided they spend no more than 90 days in any single country (see here for details).