Mobians are creatures in various Sonic the Hedgehog continuities for which the main setting is the planet Mobius (from which the name Mobian is derived). The precise definition of Mobian varies by continuity, sometimes being a general term for residents of Mobius (and used in comparison to residents of other planets, including Earth where these are different), and sometimes referring specifically to the various species of anthropomorphic animals that are the main residents of Mobius (and thus used in comparison to other species, e.g. Humans).
Mobians are sentient, capable of speech, and generally (but not universally) bipedal and anthropomorphic. Even those based on animals typically have human-like hands with five fingers. Some may also have toes, although that is less common since, for instance, Mobians who are based on anthropomorphic ungulates have hooves for feet.
Mobians in the various continuities
In Adventures of Sonic the Hedgehog, the population of Mobius consists mainly of anthropomorphic animal Mobians. However, humans feature regularly in the series in certain communities (e.g. Hill Top Village, Robotropolis).
Anthropomorphic animal Mobians are the majority of Mobius' population. Known exceptions to this rule are human-like characters (such as Dr. Robotnik, his nephew Snively, and Lazaar), dragons (including Dulcy and Sabina), dinosaurs (like the Terapods) and more monstrous characters (such as Naugus and the Kraken).
Sonic Underground depicts a Mobius inhabited principally by non-human Mobians, although the variety of forms is greater than in previous series. While there are the usual anthropomorphic animals (based on dogs, horses, etc.) there are others that bear little resemblance to standard animals, having six arms, three legs, etc. Dr. Robotnik is the only human in the series.
In Sonic the Comic, anthropomorphic animals comprise the majority of the population of Mobius, as well as some other planets (the Drakon Empire are mainly fish people) and dimensions (most notably the Special Zone). A large percentage of the population is humanoid without being based on a specific animal species (including Grimer), and there are a very small number of humans (including Dr. Kintobor/Dr. Robotnik and Zorabel).
In the Archie Comics continuity, the term mobians (written with a lower-case 'm' unlike the term in other continuities) exclusively refers to the anthropomorphic animals that live on the main planet in the setting (called Mobius before the hard reboot caused by the Super Genesis Wave and simply Sonic's World thereafter) and similar dimensions. Other species in the setting have their own names (including humans and Overlanders). Sally Acorn also referred to the species as Mobo sapiens.
The pre-Super Genesis Wave continuity describes the origin of the mobians. In this continuity, Mobius is a futuristic Earth. After a human act of aggression towards an extraterrestrial emissary sometime in the 21st century, the alien Xorda bombarded the planet with powerful gene-bombs with the intention of eradicating all humanoid life on the planet. These explosives caused the native animals of the planet to mutate and rapidly evolve, growing larger and more intelligent. When the aliens return, they reveal that the new mobian species "contain a sizable percentage of human DNA".
After Sonic & Mega Man: Worlds Collide, the planet setting of the series was renamed from Mobius to Sonic's World, and the term mobian was consequently dropped temporarily. It was later reestablished in the summer of 2016, as a colloquial term to refer to the "antropormorphic animals" that inhabit Sonic's World.
Post-Sonic Adventure games
Back during the Dreamcast era, SEGA planned to eliminate the Western and European concepts of Sonic. In the games post-Sonic Adventure, Sonic and his friends live on Earth. There is no official name to replace "mobians" so far.