There's three different types of gene loci. There's loci that make base coats, there's loci that modify base coats, and there's loci that put patterns of white on the horse.

Base Coats
The loci that make the base coats are A, E, W, and G.

First let's talk about W and G. Those two are simple, because they come in and take over the coat and cover up anything else. They're dominantely expressed, so if a horse carries it, he shows it. Modifying genes don't generally act strongly or at all on W and G. W is for White. WW or Ww makes a white horse, while ww makes a some-other-color horse. White covers up everything else.

G, the Grey locus, covers up everything except white. Grey is also dominante, so GG or Gg will be a grey horse. A grey horse starts out life showing whatever base color is underneath. As it grows, the horse's coat slowly changes to white, eventually becoming all white.

E, the chestnut (or red) locus, is also bossy, but it's recessively expressed. If the E locus has ee, then the horse will have a chestnut base coat. The cream and dun modifiers do act on chestnut. More on that in a minute. Because E is recessively expressed, a horse can be carrying chestnut without showing it (Ee).

If neither the W, G nor E locus have hijacked the coat color, then the A locus gets its chance to shine. The A locus has three different genes which can make three different coat colors. The most dominant is A, Bay, followed by At, Brown, and finally the recessive a, Black. The A locus works out as follows:
AA - Bay
AAt - Bay
Aa - Bay
AtAt - Brown
Ata - Brown
aa - Black

Coat Modifiers
Ok, that's it for the base coat. Now on to the modifiers. The two main modifiers are the Cream locus, C, and the Dun locus, D. Both act on chestnut, bay, brown and black base coats. (There are also coat-specific modifiers, which will discuss after.)

The Cream gene is dominantely expressed, but it shows differently depending on whether there's one C cream gene or two.
One C gene (Cc):
on Bay -> makes Buckskin
on Brown -> makes Sooty Buckskin
on Black -> makes Smokey Black
on Chestnut -> makes Palomino
Two C genes (CC):
on Bay -> makes Perlino
on Brown -> makes Perlino
on Black -> makes Smokey Creme
on Chestnut -> makes Cremelo

The Dun gene is also dominantly expressed, but it shows the same whether it's DD or Dd. It acts on the following base coats:
on Bay -> makes Yellow Dun
on Brown -> makes Mouse Dun
on Black -> makes Blue Dun aka Gruello
on Chestnut -> makes Red Dun
and sometimes it shows on grey for a Grey Dun.

Now let's talk about coat-specific modifiers. These are modifiers that have a specific effect on the coat colors generated only by one locus. They are the Flaxen locus, F, the Silver Dapple locus, Z, and the Fleabitten locus, B.

The Flaxen locus works only on the E locus chestnut coat base. It causes a chestnut horse to have a lighter mane, ranging from near-white to pinkish-red. If a horse is chestnut, then the Flaxen locus has a chance to show. Flaxen is recessively expressed, so a flaxen horse is ff. FF and Ff are non-flaxen. A horse with ff who is a base coat other than chestnut will not show flaxen. In other words, the F-locus can have whatever genes it wants, if the coat isn't chestnut it's not going to manage to do a thing to the mane color.

The Silver Dapple locus works only on the A locus coats, or on Bay, Brown and Black horses. It causes the mane to become lighter, similar to Flaxen, and can also cause lightening and dappling throughout the main body of the horse. Silver Dapple is domiantely expressed, so ZZ or Zz will be Silver Dapple. A horse who doesn't show an A-locus base coat cannot show Silver Dapple.

The Fleabitten locus, B, works only on the G locus grey coat base. It causes random very small patches of darker hair throughout the grey coat. Fleabitten is recessively expressed, so bb shows fleabitten whereas BB and Bb are not-fleabitten. Again fleabites show only on grey base coats.

White Patterns
Finally lets talk about pattern loci. Pattern loci are Roan (R), Appaloosa (L), Tobiano (T), and Overo (O).

Roan causes white hairs scattered randomly (about every-other hair) throughout the body of the horse, generally with less on the head and legs. Roan is dominately expressed, so RR and Rr make a roan horse.

Appaloosa causes spots and often a sort of roaning on the horse. The L gene determines whether Appaloosa shows or not. Appy is dominately expressed, so LL and Ll are appy horses, whereas ll is non-appy. Appaloosa has a companion locus, N, that determines how spots are distributed. The N locus plays out as follows:
NN - Snowflake Appaloosa
Nn - Blanket Appaloosa
nn - Leopard Appaloosa

Tobiano causes a specific paint pattern on horses. It is dominately expressed, so TT and Tt are tobiano horses, whereas tt is a non-tobiano.

Overo causes a different paint pattern on horses. Overo is also dominately expressed, so OO and Oo horses show an Overo pattern. oo is a non-Overo horse. Overo has a companion locus, P, that determines exactly what type of Overo pattern will show if Overo shows. It plays out as follows:
PP - Frame Overo
Pp - Sabino Overo
pp - Splash Overo