The two-gene model
Since 1907 a simplified version has been taught in which eye colour follows a dominance hierarchy: brown dominates green, and green dominates blue. Blue is the recessive variant: for a child to have blue eyes, they need to inherit the 'blue' version of the gene from both parents.
This explains the best-known pattern: two blue-eyed parents almost always have blue-eyed children, because both can only pass on that version of the gene. It also explains the opposite surprise: two brown-eyed parents can have a blue-eyed child, if both unknowingly carry a hidden copy of the blue allele inherited from a grandparent.
Why the result is a range, not a number
Looking at someone's eye colour does not reveal whether they carry one or two copies of the dominant allele: a brown-eyed parent can be 'pure' for brown or a hidden carrier of blue, and the two look identical from the outside. Without that information there is no honest way to give a percentage — any calculator that does is assuming a genotype it cannot actually know.
That is why this tool sorts possible colours into three tiers: likely (the most probable given the couple's known genetics), possible (occur when a hidden allele is involved) and rare (outcomes the 1907 model does not explain well, usually because more genes are involved than this simplified model accounts for).
What the 1907 model does not explain
Real eye colour does not depend on two genes but on at least sixteen, according to Sturm and Larsson's 2009 review. The main ones are OCA2 and HERC2, which regulate how much melanin is deposited in the iris, but other genes shade the result further — hazel, flecked eyes, heterochromia — none of which this classic model even attempts to predict.
It also changes with age: many babies are born with grey-blue eyes that darken over the first 6 to 12 months as melanin builds up in the iris. The final colour usually does not settle until after the first year, so a newborn's blue does not always predict an adult's blue.
What about hair?
Hair colour carries the same underlying complexity — it is also polygenic — with one notable exception: red hair. It is largely tied to a single gene, MC1R, and is recessive: it takes two copies of a 'non-functional' variant of that gene to have red hair, well documented since Valverde and colleagues' work in the 1990s.
That is why red hair can 'skip' whole generations and suddenly reappear: two parents with not a single red hair between them can both be silent carriers of one copy of that variant, and happen to match in their child. There is no reliable calculator for this without knowing the parents' MC1R genotype, so here it stays as an explanation rather than a number.