|This article is about an older version of DF.|
Extinction means one of two things in Dwarf Fortress.
 Sapient extinction
The first meaning of the word pertains to world generation: fighting between different sapient species may be so intense that a particular sapient species is wiped out entirely, going extinct in the process. Extinction events are exceedingly rare in larger worlds, but surprisingly common in pocket ones, where inter-species competition is at its fiercest. Sapient species that go extinct will obviously no longer trade with your dwarves, nor threaten them with ambushes or sieges. You can somehow still start a fortress mode game in a map vacated of any dwarves, but you will not receive any migrants beyond the first two (hard-coded) waves.
On a smaller scale, landmass-specific extinctions can also occur, and although they are not nearly so dramatic as mass extinctions, and are in fact quite common, they can still affect your game. This usually happens on separated landmasses and islands inhabited by more than one sapient species below a certain area. Since civilizations cannot yet cross bodies of water that don't freeze over, such extinctions will affect your gameplay should you chose to settle on such a landmass. The effect of settling someplace inaccessible to dwarves is the same as starting a fortress when they are extinct: only two, hard-coded, migrant waves.
Calendar ages are greatly influenced by the relative lack or abundance of sapient life in your world, and so provide an at-a-glance view of the state of civilization on your lonely planet (if you know what they stand for, of course). Two calendar ages are more directly associated with extinction: the Age of Death, in which all civilizations have been wiped out, and the Age of Emptiness, in which all sapient creatures have been wiped out (includes those that don't form civilizations, such as gnomes). The latter is an edge case: it can only be achieved by killing every thinking creature in a world in Adventurer mode, then committing suicide, then embarking on the same map in fortress mode, then committing suicide with all of those dwarves. Once all of this is done, you will finally get an announcement telling you of the passing of the age. It's been done before.
It should be noted that due to a bug, carnivorous sapient civilizations (those with the [CARNIVORE] or [BONECARN] token in their raw files) will go extinct near immediately. This is both a blessing and a curse for modders: a blessing because it allows the forceful extinction of modded civilizations, a curse because it prevents you from modding in and playing a purely carnivorous race.
 Site extinction
The second, more nuanced form of extinction has to do with the wildlife in your surroundings. Every site can have a certain number of creatures appear in it; which creatures depends on the site's biomes, how often depends on the [FREQUENCY] creature token in their raw files, how many depends on the number of biomes you abridge and on the creature's [CLUSTER_NUMBER:#], and how many before none will ever appear in the wild again depends on a somewhat randomized token in specific creatures' definitions, the [POPULATION_NUMBER:#:#] tag. That is, if as many (wild, non-siege mount) creatures die or are captured and kept upon visiting your fortress as [POPULATION_NUMBER] is set to, they will be considered locally extinct, and will have their range clipped from your fortress surroundings.
This cap will not limit any breeding you do once you have captured the creature, and will obviously not affect any creatures visiting your site that are allowed to leave. It will affect the maximum number of non-breeding creatures that you can have at your fortress, however, and will not allow the appearance of creatures in clusters larger than are allowed by the cap. Note that the cap is biome-specific: even though sea serpents are capped at one serpent per site, it is possible to get multiples, and even start a breeding program, if your fortress abridges multiple valid biomes - note that such a breeding program wouldn't work in the current version, as sea serpents are now egg-laying and aquatic egg-layers don't breed due to a bug)
Every time a species becomes locally extinct it is (obviously) taken out of the wildlife visitation rotation. Rendering a policy of mass genocide on the countryside can thus eventually force the appearance of a sought-after species of wildlife, if you push hard enough. However, for the most part the number of creatures that can appear in the biome is so large, and the population caps are set so high, that it doesn't seriously affect gameplay. Aggressively hunting the local wildlife will eventually restrict its variety, however, and eventually you can prevent any wildlife from spawning at all from spawning on your map.
The cap does have important ramifications for a few specific species, however. Wildlife appearances on glaciers are heavily restricted, for instance, so the particularly aggressive yeti, which only appears five to ten times, is often made extinct very quickly. Because of a bug, aquatic vermin do not restock when fished, resulting in guaranteed eventual extinction.Bug:2780 A certain nuance of the tag's implementation allows maintained hunting of certain species, with some micromanagement: if a cluster (or pack) of animals appears on the map, as long as at least one animal in that cluster is allowed to leave, the species will not have been considered depleted. This obviously does not work on solitary creatures.