For the most part genes come in pairs, one from mom and one from dad. From time to time mutations happen causing different versions of some genes (alleles). For each locus, if the two copies of that gene are the same it's classified as a homozygous genotype. If the two copies are different versions then the genotype for that locus is heterozygous.
So heterozygous just means having an unmatched pair of genes at whatever locus you are talking about.
Knowing genotypes is important for predicting offspring in complex breedings. Heterozygous animals have a 50/50 chance of which allele they will give each offspring while a homozygous animal will give each baby the same version 100% of the time. You can think of breeding like flipping a coin for each parent for each gene. A heterozygous parent has a two sided coin, sometimes you get heads and sometimes you get tails. A homozygous animal has a rigged coin where both sides are the same.
To know if a het looks normal or not you must know the type of the mutation you are talking about. With a co-dominant mutation like pastel the hets are a visual mutation (aka "pastel"). The homozygous co-dominant mutation type animals are a different visual mutation from the hets (aka super pastel). With a dominant mutation like pinstripe both the hets and the homozygous mutant animals are the same visible mutation (phenotype). With a recessive mutation type like albino the hets look normal and only the homozygous mutant animals show the mutation.
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