Every horse has a pair of base colour genes that determine what basic colour he will be.  There are only 2 possibilities for each of these genes.  They are either red or black.  Black is always dominant over red, so if a horse has a black gene, it will always be black based, even though it may also carry a red gene. 

Red can be thought of as the absence of black, so then, in the absence of any black gene, a horse will be red based. 

Black being the dominant gene, is shown as a capital "E", and red being recessive is then shown as a lower case "e" for colour genetics purposes.  Since each horse has a pair of genes, there are three possible combinations of the "E" and "e" genes.  A horse will be either be "EE" , "Ee", or "ee".  Only the last one "ee" would have a red coat, the others would be black based.  The "Ee" horse would be a Black based horse, but could also pass on a red gene to its' foals, so if crossed on a horse that was reb based could throw a red based foal, just as is shown in the collage of black based horses below, the black mare has a chestnut colt.

Agouti - Agouti is the modifier that produces Bay from a black based horse by restricting the black to the points, like the legs, mane, tail andear tips.  In the case of brown, which is a variation of bay, most of the coat will appear black, but brown will show on the nose and flanks.  It does not affect red bases, but can be carried by a red horse and passed to its' offspring.  Its' gene is represented by an "A" if it is present and an "a" if it is not.  Each horse has a gene pair for this, and will be "aa" - does not carry any agouti, "Aa" - carries one copy, or "AA" - carries both copies.   Whether a horse is "Aa" or "AA", makes no difference in the appearance of the horse, it will appear bay.  The shade of bay can range from light bay to nearly black. 

Cream - Cream is the modifier that creates palomino, buckskin, smokey black, cremello and perlino. Its' gene pair is represented by "C" when it is present and "c" when it is not.  So, combinations of "CC", "Cc" and "cc" are possible.  Cc means that a horse has a single dilution, and dilutes a red base to palomino, a black base with agouti (a bay) to buckskin, and can hide in a black (without agouti), making it a smokey black.  If a horse is "CC", he is said to be double dilute, which further dilutes a red to cremello, bay to perlino, while black (without agouti) remains a smokey black.   

Dun - Dun is the modifier that created duns, red duns, and grullos.  Represented by "D" or "d", with possible gene pairs "DD", "Dd", or "dd".  It is similar to the Agouti modifier, in that a double dose of the gene does not produce more of a change than a single dose.  If dun is present, a black becomes grullo, a bay becomes dun, and a red base becomes red dun. Duns are characterized by a dark dorsal stripe down their backs, zebra striping on their legs, and dark points on the ears and lower legs.  These markings are a way to visually distinguish a dun from a buckskin, while a buckskin will have the dark points, they will usually not often have the dorsal stripes and leg barring.  That said, a horse can have both the cream and dun genes, so would look dun, but genetically carry both "C" and "D".