I've heard the term used when talking about reptile morphs, and there's sometimes a percentage combined with it. What are people talking about?
First thing to know is that "Het" is short for Heterozygous.
Now, in terms of pets, in this case reptile breeding:
The first thing to remember is that it doesn't take a geneticist to become a breeder. Terms are commonly borrowed, but sometimes/usually misused. In this term here, referring to an animal as a "Het" is saying that it has an inherited trait from it's parents, but doesn't display it itself. So if a ball python was born from an albino parent, it would be and albino "Het" ball python, because even though it looks normal, it has the strain of albinism from it's parent.
This is important with breeding because in order to produce albino offspring, the trait has to exist in the parent, so with an albino "Het" there's still a chance of producing a child that displays the trait.
The numbers are meant to be the percentages of the trait estimated to be inherited by the animal from it's parents. It's used as sort of a guesstimate about how likely the children are going to receive those traits as the dominant ones. So basically if you're looking to breed for albinism, you want the highest possible chance of having the trait passed on. Likewise if you don't want to display a trait, you try to breed with the lowest chances for that trait.
I suggest you take any numbers with a grain of salt, as people and numbers don't really mix. Even if a breeder happens to be a biologist, that doesn't mean they've taken the time to crunch the numbers correctly, or even studied genetics properly. The best solution is to buy from a breeder who has been established for a while, and keeps track of their lineage. That way you can see who the parents are, what traits they display, and possibly what traits their parents display. Then you will be able to create your own guesstimate on what your animal has inherited but doesn't display.
A basic guideline that I follow, and seems fairly commonly used, is that 100% "Het" means that at least one of the parents displayed the trait visually. If it's said to be 50% Het, then one of the parents is "100% Het". Anywhere between "50% Het" and "100% Het" means that both parents were "Het" to some percentage. Any less than "50% Het" means that it's a generation or two being bred with others who don't display the trait.
Again, these are all loose numbers, people dedicate their lives to studying genetics, we're only interested in increasing chances of breeding here.
Now, in terms of biology:
Disclaimer: I am not a geneticist, so this will still be rather simplified. I would suggest heading over to biology.se for greater detail.
Zygosity is the degree of similarity of the alleles for a trait in an organism. (Source) An Allele is a form of gene, it's basically the part of the genetic structure that decides how everything is placed together. When a person has a certain color of hair, or eyes, that is decided by the Allele. More importantly, by the dominant Allele.
So, Zygosity is the measure of the Alleles that have been inherited by the organism's parent to form a certain trait. Now, some things to take into consideration, which are not necessarily relevant, but might help get an idea of how this works is:
Humans, along with most mammals are diploids. Meaning they take two copies of chromosomes, one from each parent. So pairing an 'X' chromosome with a 'Y' chromosome produces a male offspring.
Some of the earlier species such as fish and amphibians are polyploids. This is where it gets complicated and they can take three or more sets from the same, or closely related species. Ask biology.se about this one if you're interested.
Back to Zygotes, there are a couple different kinds we need to consider:
Homozygous: This is pair of identical Alleles that create a constant trait that will always be displayed. A purple flower will always produce more purple flowers (unless it's bred with a non-purple flower, and then it's not Homozygous anymore). There are two "sub-types" of Homozygous organisms:
Homozygous Dominant: Meaning that it is a pair of Alleles with a dominant trait (this would be the purple flowers).
Homozygous Recessive: Meaning that it is a pair of Alleles with a recessive trait. This would be if you bred an animal for the albinism trait to where the albinism trait was the dominant Allele.
Heterozygous: This is a pair of differing Alleles, of which only the more dominant Allele will be displayed. For example, if it is a pair of Alleles that determine hair color, and one would determine blonde, the other red, the hair color will be blonde because that is the dominant Allele. But, the Allele for the trait of red hair still exists, and can be passed on to the next child to possible be matched up with the same allele, or one that's more recessive (less dominant?).
The problem with pairing up Heterozygous Alleles, is that they can be infinitely complex (which is why people devote their lifetimes to studying them), I've simplified it to where one Allele is dominant over the other, and that's basically how it works, but determining how they are dominant, and which ones are dominant, is a complicated study that's out of my area of expertise. People devote lifetimes of effort into this area of study alone, just mapping out how Alleles are paired together and how they choose which one "wins".
Hemizygous: This one's a bit more than I can explain. From what I understand, it's when one of the Alleles gets removed, and there becomes a single trait leftover. It has something to do with the reason why men only have on 'X' and one 'Y' chromosome.
Nullizygous: When two mutations that cause a loss of function pair up. This is another one that I haven't studied specifically, I believe it's what causes defects, and/or kills the organism. Don't quote me on this.
Phenotypes: This was mentioned in a comment, so I wanted to touch on it as best I could. It's a combination of the Genotype, which are the instructions that have been inherited from the Alleles, and how it's all been influenced by the environment. Beware though, these two terms are pretty loose, and have been used interchangeably I think.
To give a gross oversimplification for an example. A phenotype would be when a moth is colored brown, because it lives in the woods and the Alleles received were for that color, and also because all the moths with different colors got eaten so the only color that was left was brown.
Matt's explanation pretty much covers the science of it. As a more simplistic view, though, consider two snakes, a male and a female, both of which look "normal". When they mate, a number of eggs will be laid.
If nothing else is known about the heritage of the two parents, then the eggs will be considered "normal", and no-one will suspect that anything other than "normal" babies will emerge.
It is possible, of course, that the two parents were both carrying the genes for an albino but no-one would have known because those genes are recessive so neither snake shows any external characteristics to give it away. In this instance, the eggs have a 50/50 chance of inheriting the albino genes from the father and a 50/50 chance of inheriting the albino genes from the mother. Statistically, we might then expect 1/4 of the babies to be normal by getting the albino gene from neither parent, and another 1/4 to be albino by inheriting the albino genes from both parents. The remaining (statistical) half of the babies will then be like their parents were - heterozygous for albino.
The babies which look normal are then known as "66% het albino", as 2/3 of them (statistically) will carry the albino gene and 1/3 will not.
That is to say, there is a 66% probability that each of them is heterozygous for albino. You'll not know, of course, until you bred them, and even then you're at the mercies of statistics so you'd never know for certain if one did not carry it, you'd only ever know for sure if it mated and produced albino offspring.