TWO DIFFERENT TYPES OF PIGMENT
The key to understanding dog genetics is simply this: there are two types of pigment which create coat colour in dogs (and most other mammals). Pigment is just the thing that gives each strand of hair its colour, just like pigment in paint or dye, or pigment in your own hair. All coat colours and patterns in dogs are created by these two pigments. Each of the pigments has a "default" colour, and it can then be changed by various genes.
EUMELANIN
Eumelanin is black pigment. All black areas on a dog are caused by cells producing eumelanin. However, there are genes which turn eumelanin into other colours - liver (brown), blue (grey), or isabella (a dusty pale brown). If a dog has any of the genes to turn its black eumelanin into liver, blue or isabella then all of the black in its coat will be changed. This is because these genes restrict and/or alter the production of eumelanin, so the cells aren't able to produce full-strength pigment. We call blue and isabella dogs "dilutes" for this reason. They either lack the genes which tell their cells how to produce proper eumelanin, or their cells know how to produce it but can't.
As well as being found in the coat, eumelanin is present in the other parts of the dog that need colour - most notably the eyes (irises) and nose. The nose will be black, liver, blue or isabella depending on the type of eumelanin the dog can produce. The colour in irises is produced by layers of pigment, and brown eyes like most dogs have (and like many people have too) are caused by black eumelanin in those layers. When a dog has altered/restricted production of eumelanin, the irises are also unable to produce full-strength eumelanin. This means that the dark pigment in the eye becomes lighter, and the eyes turn into a light brown colour, known as amber or gold. This isn't really a colour we find in human irises, so it's hard to illustrate it with an example. The closest we have are hazel eyes, which are a mixture of green and brown, and so appear lighter than normal brown eyes. Dogs, however, cannot produce that green colour, only brown and blue.
This shows the variation in colour of eumelanin. Black is the default, then it can be turned into liver by one set of genes, and black and liver
can be turned into blue or isabella by another set of genes. Blue is diluted black and isabella is diluted liver.
When we talk of dogs that are "black pigmented", "liver pigmented", etc, we mean that is the colour of eumelanin that the dog can produce. Sometimes these dogs have no eumelanin at all in their coats (their skin cells produce only the other type of pigment, phaeomelanin), but we can tell what their "pigment colour" is by looking at their nose. A black nose means the dog produces black eumelanin, and so on. It's confusing to talk of a dog's "pigment colour" like this, because as we know, eumelanin isn't the only type of pigment. But it's common practice, and I'll be using such terms a bit on this site.
PHAEOMELANIN
The second type of pigment, less important than eumelanin, is phaeomelanin. This is red pigment. The term "red" covers everything from deep red (like Irish Setters) to light cream, encompassing gold, yellow and orange. Whenever we talk of red, unless we're talking particularly about Setters, we mean the whole range of tan colours.
Phaeomelanin is produced only in the coat. It does not occur in the eyes or the nose, so any genes which affect the colour/intensity of phaeomelanin will not affect the eyes or nose. Only eumelanin occurs in those areas, and so only genes which affect eumelanin can affect the eye or nose colour.
This shows the variation in colour of phaeomelanin. Unlike eumelanin, it doesn't occur in two distinct colours (black and liver, with dilutes counting as shades of those), but rather just one colour, which
varies in intensity. The most intense phaeomelanin colour is Irish Setter red. The default colour is probably golden, with different genes causing it to be
more or less intense (i.e. telling the cells to produce a higher amount of pigment particles, so making the colour stronger, or a lower amount, making
the colour weaker, so lighter).
So far so good, but this doesn't seem to explain all the coat colours in dogs - how about white?
WHITE
White isn't really a colour, so white hair on animals isn't caused by pigment but a lack of pigment. It is a lack of both eumelanin and phaeomelanin. White areas on animals are simply caused when the cells cannot produce any pigment at all. Sometimes the whole animal is affected, like in albinos, and sometimes just parts of it are affected, like in dogs with white markings. It can affect the production of eumelanin in eyes and noses too, turning noses pink and eyes blue (or red in proper albinos). There is also a second type of white, which is caused by a gene called chinchilla. The chinchilla gene dilutes red (phaeomelanin) pigment, making the cells produce less pigment particles than normal, so the colour gets lighter. If it is diluted enough, it can become white. Many white dogs have a slight ivory/cream sheen to their coats because their cells are still producing a very small amount of pigment. This sort of white does not affect eumelanin, so any black/liver/blue/isabella areas on the coat will stay dark, and the eyes and nose will do too.
DISTRIBUTION OF PIGMENT
The colour genes in dogs do two things - they determine the eumelanin and phaeomelanin colours/shades, and they control the distribution of these two pigments. They tell certain cells to produce eumelanin, others to produce phaeomelanin, and sometimes they tell them to not produce pigment at all. Exactly which cells are told to produce what is determined by the exact set of genes, although it can be random to a certain degree (e.g. puppies may have slightly different white markings to their parents, or patches in different places). Sometimes genes can even tell cells to switch which type of pigment they are producing every once in a while. This means that as a hair grows, it becomes banded with black and red, because the cell produces black (eumelanin) for a while, then changes to red (phaeomelanin), then back to black, etc. It's a bit like when you highlight your hair and after a while the roots start to show through. The overall colour of an animal with this sort of black and red banding will generally be a muddy brown from a distance, and close up you will be able to see the black parts of the hairs. It's called agouti, and it's the colour of wild rabbits and mice, as well as a large amount of other mammals. It's popular amongst wild animals because it provides very good camouflage. It also occurs in dogs, but it looks a bit different (like the colour of a wild wolf rather than a rabbit) and isn't very common.
All of this probably seems a bit confusing at this point. Many genetics sites gloss over the types of pigment for precisely this reason. However, I've decided to include this information because it really is crucial to your understanding of dog genetics (and the genetics of other animals too). Once you've grasped the idea of the two types of pigment, and understand how they work, the rest should be plain sailing.