Q:

Genetic structure of a trait with six discrete phenotypes

In my world magic is conferred genetically. Their are five "types" of magic, which never mix. You either have one type, or you have another (or none at all, the sixth option), you never have a mix of two.
In real life we see this behavior frequently with two discrete options. By that I mean two distinct options, either one or the other, not a mix or a gradation. For example, we have the green or yellow pods of the famous Mendalian peas. They are either green or yellow, and don't mix between the two. However, for most traits with a lot of options, we have a gradation, like skin color or height, and not discrete phenotypes. 
What would the genetics look like for six phenotypes rather than two?
Assume that inheritance works exactly like it does for humans on Earth. No magic is involved in reproduction or inheritance. For this reason, this question is science-based.

A:

Blood type, there are six alleles for blood type with four phenotypes. Just use a similar plan and make them all co-dominant instead the normal setup (O does code for a protein it just doesn't work). So now you have six genotypes and 6 phenotypes. 
basically take this chart and ignore the blood type column.
so A+O would be its own thing and not the same as A+A.

this also gives you a frequency you can map. So some types will be more common than others. 
If those genes are not the end result but instead code for the activation protein for different pathways, then cofactors don't even have to be related to each other. 
AA = type 1
AB = BA = type 2
BB = type 3
AO = OA = type 4
BO = OB = type 5
OO = type 6
Of course you could have the super rare chimera individual who expresses all three and thus is something else, because they are not actually one person biologically. Biological chimera are extremely rare and are a product of zygote fusion. These may occur no matter how you decide to establish the genetics. You can also just ignore these individuals and say they self destruct in the womb. 

A:

All of the answers so far either do some hand-waving with DNA bases or amino acid codes, or try to construct six discrete phenotypes from a mixture of different alleles. I'd like to offer / describe a biologically possible solution, that is at the same time very 'basic': You can have five different variants of a genes (or six with one being magically 'inactive'), one for each type of magic, but still have only one active in each person. 
The biological trick for this is called allelic exclusion. In humans the body uses it to suppress one of the two X-chromosomes in females (they only need one) and - more to the point -  to prevent certain specific genes from being expressed from two alleles. (This is important in the immune system, where any given B-cell should only produce exactly one version of the B-cell receptor/antibody).
In your case you would have six different possible alleles. While each person will have two different ones in their genome, only one will be active in each person. This also means that children can inherit a magic type (including 'no magic') that their parents are not actively showing, as long as they carry the genes for it.
Edit, a few more things:

This system works both with the 6 gene variant (-> no magic has a gene) and with the 5 variant (-> no magic is simply the absence of any gene). The latter version will mean that magic is always dominant over 'no-magic', because persons with only one gene, will obviously express it.
The silencing of gene normally has to happen in the 1-cell embryo stage, otherwise you can get people with one half of their cells 'fire' magic and the other half 'water' (spotted colouring of cats works that way). Of course, if you wanted to, some freak of nature like this could still exist with two different half powers
If anyone ever asks for an evolutionary reason to have allelic exclusion on the magic gene: maybe having two competing kinds of magic in a single cell just mean bad things for that organism. 

A:

You are in luck! Six variations fits quite nicely into the genetic code; look here at the DNA to Protein Table. 
There are four DNA 'letters', ATCG. It takes 3 letters to specify a code for one of our amino acids; or one of the control codes ('Start', 'Stop'). That is 22 results for 64 possible combinations ($4^3$) so 42 of the 64 codes must duplicate some result.
The 'winner' for this is Leucine, which has SIX distinct genetic codes: TTA, TTG, CTA, CTT, CTC, CTG. 
Proteins are a link of such amino acids that fold into some specific sequence. IRL of course, Leucine is Leucine is Leucine; the ribosome translates all six of these things into the same amino acid.
However, for fictional magic purposes, the construction of your magical organ must depend upon DNA, and you make which type of magic a person can do depend upon seeing the correct variant of Leucine in one or more places within this organ.
If you want the magic to be fairly common; this could be in one place. If you want it to be more rare, demand it in multiple places: Say the chances of one variant appearing is 1/6, then getting the same variant in two places is 1/36, getting the same variant in 3 places is 1/216, etc. People with a mixture have no magic.
This is likely how you would ensure heritable magic (or none at all), the chances of 2 nucleotides mutating exactly the same way in an offspring are very remote. However, you could have SNP (single nucleotide polymorphism) mutations (just one nucleotide changed) with a very low % chance that make the offspring of two same-magic parents have mixed leucine types and thus no magic at all.

