Quirkless people exist as a minority in the My Hero Academia universe, possessing absolutely no Quirk, particularly unique features, or anything that would put them under the classification of being superhuman. They appear to be the subject of mockery and a level of prejudice as shown by Izuku Midoriya's treatment in the first few chapters of the story. Commonly, anyone who does not possess a Quirk is considered to be weak and worthless, or that their circumstances are incredibly unfortunate and rare.
As seen in the beginning of the series, people who are born with Quirks do not have an additional joint in their small toe, suggesting a close chromosomal linkage between the extra toe joint and a lack of a Quirk; people with the additional toe joint are almost always Quirkless.
Quirkless people make up approximately 20% of the world, and the trait of Quirklessness appears to be recessive; meaning if at least one parent has a Quirk, that child is more likely to be born with a Quirk even if the other parent is Quirkless. However, if both parents are carriers for Quirks, there is a chance their child will be born without a Quirk, as seen with Izuku who inherited neither of his parents' Quirks.
It is possible for a Quirkless person to gain a Quirk later in life, such as One For All. However, they must train their bodies in order to be able to use the Quirk, similar to how Izuku worked hard to earn his Quirk. This is because, unlike other people who are born with adaptations, they are born with none.
Currently, only three people in the entire franchise are confirmed to have been born totally Quirkless: Izuku Midoriya, Toshinori Yagi, and Melissa Shield. However, it is possible to steal a Quirk, as seen with All For One who rendered Tomoko Shiretoko Quirkless, despite having been born with a Quirk. Quirks can be destroyed in a particular way; currently only the Yakuza villain group, the Eight Precepts of Death, managed to achieve this by using Quirk Destroying Bullets. Using these bullets they rendered a third-year student at U.A. High School, Mirio Togata, Quirkless. Knuckleduster has also been rendered Quirkless by unknown means. However, it is believed there is a way to restore one's Quirk. Shota Aizawa believes that if the user of the Rewind Quirk masters it enough, they could restore lost Quirks.