In Warmblood breeds the most frequently appearing color is all kinds of bay:
The lightest bay will be called hellbraun followed by simple braun, dunkelbraun. Horses called schwarzbraun are nearly black.
Chestnut is called Fuchs. It is devided similar to the US. Some of the differences are rarely used and there often exist different opinions about what should be called e.g. liver chestnut. The most common kinds of chestnut color are (in order of brightness Hellfuchs, Fuchs, Dunkelfuchs (also known as Kohlfuchs) which can be nearly black like Donnerhall.
Any horse that is totally black will be registered as a Rappe. Since the foals are mostly not yet black when the paperwork on their first registration is done, it sometimes happens they are primarily called schwarzbraun and later the color is changed into Rappe after it has been inspected again.
Now for the grey: They're called Schimmel in German. Roan is divided into Rappschimmel, Braunschimmel or Fuchsschimmel depending on the base color. There are some more specifications of grey such as Apfelschimmel (dappled grey), Fliegenschimmel (white with sparkles of brown on it) or even Silberschimmel (silver) that are not officially used but sound elegant.
Horses born white are called Albino and similar to other albino animals in danger to catch sunburns or even more serious diseases of the skin or eyes. That's why this "color" is not wanted in German warmblood breeds.
Piebald is called Rappschecke and skewbald will be described as Fuchsschecke. Spotted horses that have a white coat coined with dark spots like the Danish 'Knabstrupper' and the horse of Pippi Langstrumpf of Sweden and Dalmatian dogs are Tigerschecken. Like these dogs they share the riscs of albinos due to a lack of pigment depending on the rate of colored hair.
Colors like tobiano, overo, known as everyday colors in some US western horse breeds are rather rare among original German breeds. They are either the result of fate or of the efforts of a small number of stallions owners and breeders who really have the goal to produce a specific purebred color. So there is no German vocabulary for these colors. Palominos are sometimes mentioned as Isabell other than original Haflingers who are looking alike but call themselves Fuchs mit weissem Behang.
Duns are declared simply as Falbe not minding the different kinds (e.g. buckskin).
Please notice that in Germany the colors are normally used as nouns. E.g. one wouldn't say: This horse is grey, chestnut or black. Germans will always say: This horse is a Schimmel, a Fuchs or a Rappe. Talking about two or more horses might confuse you a little: It goes 'two Schimmel', but 'two Rappen' but don't worry. Noone will call you dumb if you don't know our crazy grammar. If you're unsure about the plural just ad an 's'. Although that is not correct anyone will understand the meaning, so it might be useful in everyday life.
The best comes up right now: Talking about two chestnuts you probably have a first look at our 'ü', one of the three special vocals making our language a mystery to people from foreign countries like the 'ø' and others in scandinavian countries. Want some more? Pferdeanhänger (pronounciation like in 'headbanger') means Horsetrailer, König (try to form an 'o' and an 'e' like in 'pen' at the same time) means king.
Plural of one Fuchs is two Füchse. In case there is no 'ü' on your keybord another correct way to type it is 'ue'. To give you an idea of the pronounciation: Try to mix up the vocals of 'fee' and 'poor' in one sound. You can also look out for the next Turkish TV-show and listen carefully, there are plenty of 'ü's in them.