They do that in a hotel, hostel or hostel, a private home (commercially, i.e. a bed and breakfast or guest house place, vacation rental, or non-commercially, with members of hospitality services or in the home of friends), in a tent, caravan/camper (often on a campsite). In addition there are make-shift solutions.

Sleeping is typically done lying in a bed, or more generally on a soft surface, such as also an air mattress, a couch, etc. Some trains have sleeping cars.

Sometimes people sleep sitting, because lying is not possible, e.g. in a train (if not in a sleeping car), a bus, a seat in a waiting room, a bench on the street or in a park, etc. Inclinable seats allow something between sitting and lying. Whether lying on a row of seats is possible and comfortable depends e.g. on the presence of arm rests, and whether they can be moved up. In some public places lying would be possible but is not permitted.

Make use of the internet to search for hotels in the area you plan to visit. Use search engines, lodging directory, travel Websites to find affordable hotels.