That is actually a very interesting question and an example of how what seems like a simple concept can have different definitions depending on context. Biologically speaking, a mutant can be defined simply as any creature with a mutation, but that definition does not help us categorize the world into "mutants" and "non-mutants" because everyone has some sort of mutation. Socially, I would define a "mutant" as someone who has one of the following:
1) A significantly above-average presence of mutations. In other words, this person is much more mutated than the average person in their family or community.
2) A mutation that has an unusually large effect on the person's life. For example, an Ethiopian who spontaneously developed a mutation for red hair and became a celebrity and research subject just for that reason could be called a "mutant".
3) A mutation that was artificially induced, e.g. via intentional genetic engineering or accidental exposure to environmental toxins, research chemicals, nuclear waste, and other "mutagens". This covers your basic "mutants" as they appear in science fiction stories.