Accidents happen, but when that accident involves exotic matter or radiation, that's how mutants are born. But not every individual exposed to dangerous hazards becomes a mutant, in fact, most become dead. A mutant is someone exposed to dangerous radiation, matter, chemicals, what have you, who also has an inactive, latent powered gene. The abilities of a mutant vary wildly, they vary not only on the mutagen that they were exposed to, two people could both shoot energy from their hand where one is radioactive while the other is electromagnetic. It also varies on the individual's genetic predispositions as well as their personality, so two people exposed to the same exotic matter could develop super speed while the other gets super strength depending on how each person acts.
Mutants tend to be few and far between, with only one person developing powers at a time, though there have been some instances where many people gained powers at once. Some of the first recorded mutagenetic instances were from particle accelerator mishaps in the early 20th century, as well as nuclear and atomic weapons testing in the same era. One of the largest mutagenic incidents was the Hope Creek Accident in the 80s, though it's thought that the street drug known as Alice may potentially be a cause for mutations.