The city was founded in the 10th century and named after Thor, the god of thunder and lightning in Norse mythology. In the early history of the Faroes Tórshavn became the political center of the Faroes and still is today. The first settlers established their own parliament called "ting" around 800. The main thing was established on Tinganes in Tórshavn and local things where established in different parts of the islands. Tinganes drew people to trade and to attend court meetings etc. Currently the city has a population of around 20,000.
City buses operate 4 routes which reach most of the city. All city buses are free to ride. They run every 30 minutes from the city centre on weekdays, hourly on Sunday and Saturday and weekday evenings. There is no bus service during Saturday or Sunday evenings. 
View to the ferry port of Tórshavn, seen from Tinganes, which is the old part of town.
Faroese løgting (The Faronese parliament), Tinganes. An impressive dark red structure with a green roof located on the water front. The parliament has been meeting at this site since 900 AD.
Tórshavn Cathedral. An attractive white building with clock tower - established in 1788.
The Nordic House. The main cultural institution on the islands.
The Historical Museum. Well worth a visit.
Faroese Music Gallery See, feel and hear Faroese music in the the newest and most popular tourist attraction. Free live concerts Mondays to Fridays and on special occasions (see website for program) from 14th of June to 13th of August and free coffee.
There are many shops in Tórshavn, especially down town and in the shopping mall SMS.
Brasserie Hvonn, Hotel Tórshavn, Tórsgøta 4, +298 350000, central location and non smoking inside, American style menu and salad bar
Restaurant Glasstovan, Hotel Føroyar, Oyggjarvegur 45, +298 31 75 00, Located on a hill above Tórshavn with great views (if there is no fog), non smoking, buffet on Mondays evenings with traditional Faroese foods.
Burger King, SMS Shopping Center, +298 32 34 20, a burger restaurant.
Sunset Boulevard, SMS Shopping Center +298 32 34 22, a danish chain of sandwich restaurants located in the SMS Shopping Mall.
Baresso Coffee, SMS Shopping Center +298 23 34 32, a danish chain of cafés.
MIMIR, Varðagøta 2 (There's no sign but it's pretty obvious that it's a pub from the outside.), ☎ 31 37 48 / 28 37 48. An eclectic pub if there ever was one. This signless watering hole attracts a mixed, friendly crowd and has reasonably priced (35 kr) beers on tap. On the bottom floor there are pool tables.
Tórshavn Camping, Yviri við Strond (It's about 1km from the harbor. Bear to the right as you exit and follow the seaside (Yviri við Strond) - it's on the right.), ☎ (+298) 302425, . Pretty low-frills campsite catering to both tenters and trailers. There's a spartan kitchen, (free) hot showers plus coin-operated laundry facilities. A convenient walk to town and not too far from a large supermarket further up the road to the northwest.70 DKK / tent.
Prices vary slightly from around dkk 250 to dkk 300 per night/person for adults.