Gestures are a type of inventory item that trigger your avatar to animate, play sounds, and/or emit text chat. Created with a series of steps, gestures can be used for practical purposes or to amuse friends. Gestures can make it easy to repeat long phrases that you need to type frequently; they are also used in combat to make it easier to use weapons. Their uses are almost limitless1).
Note that it is not advisable to have too many active gestures. In some rare cases, this can lead to problems logging in; please see here for more on that.
Your gestures are generally saved in your Inventory, in the Gestures folder, but they can be anywhere. Some are supplied with avatars, or with weapons, and so will be placed in the same folders. To conveniently see which gestures you have active, you can click the Gestures button on the bottom button bar, or you can go to the top menu → Comm → Gestures, or, simply press Ctrl-G. These last two open the Gesture List Window.
You can create your own gestures too. Right click any folder name and select New Gesture. This will open the Gesture Editor. Similarly, gestures for which you have modify permissions can be edited, by right clicking them and selecting Open.
For much more information on gestures, refer to this SL wiki page.