Several night markets (夜市) are located in each district. Some are open during daytime, and all are open until around midnight. Night markets consists of restaurants and stores at the permanent locations and little booths along the center. Every night market has a huge variety of food, so any night market you find is a good bet for good food. Because of the vast selection, the recommendation is to go with a few people and share the food. Vendor food is generally safe to eat, but use common sense though if you have a sensitive stomach!
The most famous one in Taipei is the Shilin Night Market (士林夜市). It is easily accessible via the MRT (red line) at either the Jiantan (劍潭) or Shilin (士林) stations. It has a large game section, and many food shops. Nearby are vendors, and haggling is appropriate here for non-food vendors, especially clothing. Also, they may raise prices for you if you don't look Taiwanese. This is known as the most "touristy" night market, and shop owners generally can speak English.
Locals in Taipei view Shilin as touristy, with food catering to the tastes of mainland visitors. Another excellent option is Ning Xia Yeshi.
Gongguan Yeshi is located on the western side of Roosevelt Road across the street from National Taiwan University (Taida). When exiting Gongguan station (green line), use exit #1 and follow the sidewalk until you see stands lining the side-streets. This night market is closed on Wednesday nights. It is known for its variety of food, and for most shop owners knowing English due to the international student population at NTU.
Shida Yeshi is a clothing-focused night market. Some food is available, but with limited options.
Ximen Yeshi is not technically a true night market, but is a very lively shopping area come sundown. It is one of the most modern markets in Taiwan, and there is often an abundance of acrobats and other street performers. It is located at Ximen Station off the MRT green line.
Longshan Yeshi is a classic part of Taipei located at Longshan Station on the MRT blue line. It is home to "Snake Alley" and the famous Longshan Temple, which is one of the largest temples in Asia. Taiwan's "red light district" is located nearby, and a number of the city's homeless congregate here due to free food offerings from the temple. However, crime is extremely low so you should not be worried!
Raohe Yeshi is one of the "most Taiwanese" night markets within a few KM of downtown. It is one long road, and is known for its local foods and cheap socks.
Danshui (Tamsui) Yeshi is located at the farthest stop on the MRT, Tamsui Station at the northern end of the red line. It is known for its beautiful view along the river, its massive ice cream cones, and its fresh, affordable seafood.
Miaokou Yeshi is located in New Taipei City, specifically Jilong (Keelung). It is right next to the Keelung Harbor, and is known for its extremely diverse food offerings, including flavored ice, fried dough, and fresh seafood.
Lehua Yeshi is located off the MRT yellow line. It is in very Taiwanese territory. Few shop owners speak English, and the food is notably local.
Some of the best known night market snacks are:
Oyster omelet蚵仔煎; ô-á-chian
Tianbula甜不辣; tiánbúlàLiterally "Sweet, not spicy", is a Taiwanese version of Tempura.
Stinky tofu 臭豆腐; chòudòufǔ
Mango ice芒果冰; mángguǒbīng
Pan fried pork buns水煎包; shuǐjiānbāo
Taiwanese sausage香腸; xiāngcháng
Pearl milk tea珍珠奶茶; zhēnzhū nǎicháa classic drink invented by a tea vendor in Taichung.
Braised soy bean and tea eggs滷豆乾&茶葉蛋; lǔdòugān, cháyèdànbraised soy bean and tea eggs