Skin and hair pigment is made up of different types of melanin. There are two broad groups of melanin: eumelanin, which is brown, and phaeomelanin, which is red. Hair colour is a mixture of how much eumelanin and phomelanin is in your hair.
Humans usually end up with very little phomelanin because of the product of a gene called MC1R ( the melanocortin 1 receptor). The MC1R gene lets the conversion of phomelanin into eumelanin happen. If your MC1R gene mutates in a certain way, the conversion from phomelanin into eumalanin doesn’t happen anymore, leading to a buildup of phomelanin, which results in red hair (as well as fair skin and freckles). If only one of your MC1R genes is mutated, and not the other, then you have an increased chance of having red hair but you’ll probably just end up with freckles and being more sun sensitive than the average person.
So, roughly speaking, if someone has predominantly eumelanin, they would have dark hair. Somebody with very bright red hair will have little eumelanin but lots of phaeomelanin. People with auburn hair will have some of both, strawberry blonde is a little of each.
If one of the parents of a child has bright red hair (and therefore carries two of the changes – one on each of their chromosomes), and the other parent is a carrier, then perhaps 50% of the children might have red hair. And the first (mutant) redheads probably walked the earth around twenty thousand years ago.
(Thanks to the various genetics websites from which I cobbled together this explanation. Apologies to any geneticists who happen to read this and are offended by the simplifications involved.)