An Event is a term used to describe what can be thought of as a side-quest in Diablo III. However, they are not really quests, and have some fairly large differences. Events are not to be confused with Adventures.
Events vs. Quests
Many events will involve the player in performing an action for an NPC or otherwise give the player a goal to achieve, however, this is not always the case. A quest almost always involves an NPC guiding the player through the story of the game, and they are not optional. Events are optional.
Events are randomized. They take place within random tiles within any map, and are not guaranteed to spawn within a map. Some events are more rare than others. Some events do not even involve combat, but may simply be a different piece of landscape, such as the pond in the Fields of Misery. When an event does contain an action a player must take, the player is rewarded within the tile of the event. So the event is, in essence, self-contained. A player will never complete an event and have to "hand it in" to an NPC in town.
However, events often act as a sort of thematic counterpart to many quests. As an example, when the player is exploring the Defiled Crypt in the search of the crown of Leoric, they have the opportunity to participate in the Matriarch's Bones event, which is themed not only after the dungeon but the themes within the narrative of the act as well. While events are separate from quests, they often go together or "flow" nicely.
Events are not static. They can take the form of many different types and styles of gameplay and content. In each map, including randomly-generated indoor and dungeon maps, there are certain areas that are left to be always dynamic where "tiles" are cut out, and it is in these areas that events and adventures will occur. So even the over-land maps, which are mostly static, still have a random element to them, which is key to the Diablo series.
However, Blizzard took this a step further, and inserted randomness into the randomness. In the Fields of Misery, for instance, the player may spawn an event area known as the Forlorn Farm. This is best thought of as a stage, and the actions the player will perform as plays. The stage will almost always be the same--the forlorn farm--but the play being performed may be different, depending on which spawn the player gets, if the farm even spawns at all. One game, the farm may spawn a farmer besieged by monsters corrupting his fields, and the next the farmer may be dead with a dungeon entrance to a cellar. So there is randomness within randomness.
In addition to that, as mentioned above, sometimes events don't even involve combat. Sometimes they are simply a landscape tile, like the pond in the Fields of Misery, or the smoldering ruins in the Northern Highlands. Some event tile are extremely large, taking up to a fifth of the entire zone. Some events are extremely small. It is all random and incredibly varied.
Events by Act and Area
Warning yet again: spoilers below.
The main header is the zone the event takes place in, and then each event in the zone is listed.
This list is under heavy construction, and placeholders may be used.
|Quests & Events of Diablo III  Act I Act II Act III & Act IV Act V|
|Act II Quests||Act II Events |
|Quests Act III||Events Act III||Quests Act IV |