Danmaku (弾幕, literally bullet curtain or curtain fire) is a Japanese term for "barrage", as in a barrage of bullets.
Danmaku shooting games are similar to regular shooting games, but focus more on weaving through complex patterns containing anywhere from dozens to hundreds of bullets.
The comparison often used is that bullets in traditional shooters are aimed at the player, fast, but relatively infrequent. Bullets in danmaku games tend to be slower, but much greater in numbers: depending on the attacks, enemy bullets can be scattered all over the screen, covering the major part of the playing field. They aren't necessarily aimed at the player, and can be aligned indiscriminately, or in movement-restrictive patterns. The patterns themselves can be intricate and complex, giving the barrage a beautiful, if deadly, nature. The danmaku genre has attracted many fans who enjoy either the inherent difficulty, the beauty of the enemy attacks, or both.
See General Strategy for a bit of general advice for playing this type of game.
The term Danmaku shooting game has an equivalent English term popular in some circles: Bullet hell. There is also the term Manic shooting game, which can be used synonymously with the previous, although some shmup enthusiasts insist that these two terms denote different shooting game subsets that don't always overlap.
In this context, "bullet hell" denotes a high density of enemy fire that doesn't always require one to move or react fast, but rather to be calm and precise in order to navigate the bullet mazes efficiently. The examples of such games are the Touhou games, Mushihime-sama, Espgaluda, DeathSmiles (all three by Cave), etc..
"Manic", on the other hand, denotes high gameplay dynamics where reaction times are rewarded (as opposed to slow/methodic dodging) due to extremely frantic nature of the games. The examples are Raiden Fighters series (Seibu Kaihatsu), Dangun Feveron (Cave), Battle Garegga (Raizing).
There are numerous games that fully adopt both approaches, though, like DoDonPachi DaiOuJou (also by Cave). Same could be said about some of the Extra stages in Touhou games.