A trailer is a short video used to advertise and showcase an upcoming product. A trailer for a video game may or may not contain actual footage from the game in question. There have been trailers for most Halo media.
An announcement trailer is often used by gaming companies to introduce upcoming games to the public. These trailers are often pre-rendered (meaning they do not show actual gameplay footage, but rather a cinematic) and filled with action in order to excite the fans and stimulate those yet to become fans. The most common choice of where to release these videos is at E3.
Note that these trailers are sometimes created in the midst of game development, so, quite often, footage seen in the trailer is modified in the final game, or changed altogether. Because of this, it is sometimes difficult to consider the information in the announcement trailer as canon.
A gameplay trailer primarily showcases in-game or gameplay footage of a game.
A cinematic trailer can showcase a cinematic or cinematics that actually appear in the game, or alternatively, the trailer can be an entirely separate, pre-rendered cinematic of its own.