Pokémon typically fall within a set margin determined by species. Every member of a species has the same list of potential moves, same strengths, same weaknesses, and falls with a similar range in size and stats. The appearance may have minute variations, detectable only to a human very familiar with the species or specimen.
As noted by the emphasis on typical, this is not always the case.
The most common Variation is moves. Many Pokémon have been found using moves that the other 99% of their species will never be able to learn. Some of these moves make a degree of sense, such as an Entei using Ho-oh's Sacred Fire. Others are weird, like the famous Surfing Rhydon.
These non-standard moves are considered legal in League battles, but may be illegal in other formats, such as Battle Frontier.
Less common, but not unheard of, are more notable physical differences. The 'Shiny' characteristic – named for the iridescent quality of their blood, rather than the overall appearance – is the most well-known and common example (the one-gene mutation that causes it and the different coloration is in fact becoming more common, appearing in 1 in ~4000 cases rather than the previous 1 in ~8000). Pokémon that always grow their fur in rough patches, spiky patches, or fluffy masses have been found. Pokémon with certain elements of their appearance vastly different from the norm, such as a Fire Type burning with fires of abnormal color, or extra digits on paws. Sometimes, environmental factors or genetic factors trigger substantial growth leading to Pokémon more than 50% larger than normal, referred to as the Giant phenotype. Finding a rare phenotype is almost treated as a sport in itself by some trainers.
Every Pokémon has some stories surrounding them. Sometimes, that story may be limited to a village that worships Meowth, or a cave said to contain a Gengar King. Other Pokémon have legends told of them across entire regions. An unverifiable, but persistent story tells of a Dragoite the size of a skyscraper visiting several coastlines that serves as the inspiration for the Tyranilla film franchise. Arcanine is called the 'Legendary' Pokémon because of a historically persistent story that pops up in every region with evidence of its presence in the fossil record.
Pokémon that made themselves separate from the rest of their kind through action, power, or physical differences get labeled Distinct Pokémon. These are the ones the legends and myths are written about. These unique examples can create a stereotype about all members of a species, such as the idea that Spiritomb is 108 wicked spirits (It is 108 spirits, but most examples are not wicked. Only one example was made as a punishment for a wicked bandit clan).
The Type Royals are legendary specimens of various non-Legendary species that have grown to an average of 10 times larger than ordinary for their species. These examples act as guardian spirits and rulers for their Type across the whole region.