Assuming you understand the basics of genetics (genes/alleles)
All horses are either black or red.
There is the agouti gene that can only be seen on black horses since it restricts the black to the points making bay or brown. (On red there is no black to restrict)
Those are your 4 basic colors.
Then you have the modifiers. They change the color into something else.
Some are stronger with two genes. For example. Dun with change any color into color + dun. ("Dun" being the traditional bay + dun, then "Red Dun" for chesnut + dun the confusing part is often different colors have several names or people mistake the color for another, or you can have several modifiers on one horse! "dunalino" dun + palomino (cream + chestnut) but that's getting ahead of ourselves..)
Cream is an example of a modifier that is stronger with two copies.
black + cream = smokey black + another cream (black with two cream genes) = smokey cream
bay/brown = buckskin = perlino
chestnut = palomino = cremello
Interesting how some "normal" colors are so closely related to "rare" colors.
There are several modifiers, there are also modifiers that effect only a certain color (similar to agouti on black) for ex, silver will only show on a black horse. Or there are other modifiers (aside from the main ones, more specific) such as flaxen that only show on red (you don't hear "flaxen black" do you?)
I would say the common ones are cream and dun, occasionally silver depending on the breed. Others are rather recently discovered and not completely understood (though you can argue we don't completely understand a lot of science!).
There is grey, which acts like a blanket, and will slowly turn whatever color horse white as it ages. (If the horse was black it is still genetically a black horse! The genetic color NEVER changes! It is a black horse with one or two grey genes).
Then of course there is the leopard pattern (Appaloosas) and pinto patterns, but that is a whole other ball game and I think not what you were asking. Also, you need to learn to stand before you run
Love this book: Horse Colour Explained by Jeanette Gower | eBay
(I don't remember what I paid but it was NOT $60! I'm sure you can find it elsewhere.) Morgan Colors- Smoky Black Morgan Horses Equine Color Genetics
Some good for basics websites.