Touhou (pronounced as "toh-hoh", meaning Eastern or Oriental) is a series of Japanese vertical shoot-em-up games developed by Team Shanghai Alice. The setting is a fictional part of Japan called Gensokyo, an isolated island separated by a magical barrier and is teeming with youkai ("yoh-kai", Japanese supernatural creatures in Shintoism) whom the main characters do battle with. Touhou belongs to the danmaku or "Bullet Hell" subgenre of shoot-em-ups.
Bullet Hell games are characterized by an inordinate amount of bullets that the player must dodge. In other words, the objective of the game is to survive. Bullet Hells are known to feature elaborate and extremely pretty patterns of bullet flows that will eventually hurt your eyes. The player must dodge hundreds, sometimes thousands, of bullets, forcing the player to memorize patterns (because they will die at some point unless they're that skilled), skill, and most importantly, any skills they have in dodging.
Touhou is one of the few Bullet Hells where the playable characters are magical girls in frilly dresses rather than spaceships or war vehicles.