In the Twilight Saga, vampire eye colour plays an important part in telling what type of vampire the individual is likely to be. There are three main types of vampiric eye colour: red, golden and black; and these are all achieved by what state of hunger said vampire is in. You can find examples and explanations below.
 Red eyes
There are a couple of reasons a vampire would have red eyes. The most commonly thought of which, would be that they have recently drank the blood of a human. For example, vampires that live on human blood include Aro, from the Volturi, Laurent (as seen in Twilight and New Moon) and Victoria. He has a constant red coloured iris, except from when he is thirsty (see below).
There is another possibility, however. Their eyes could also be red in colour because they have only just become a vampire (they are new-born). When a new vampire is "born" (also known as created), their eyes are always red, even if they start off on the vegetarian diet (again, see below). The eyes of a new-born are always red because of the amount of and presence of human blood in their body - Bella Cullen would be an example.
It is thought that human blood makes vampires very irrational and hot headed. They usually stay alone, or in pairs and are rarely found in covens, however, though, vegetarian vampires are often found in covens, and this is thought to be because of the absence of human blood as it makes them easier to 'blend in' with the humans.
 Golden eyes
Although most vampires drink human blood, when one has golden eyes, it means that they don't. Golden-eyed vampires drink the blood of animals - the Cullens coined the term vegetarians, which means exactly this.
Vegetarian vampires are usually well mannered and can handle being around humans for a long time, but find it extremely hard to cope when their blood is exposed, they have tasted it recently or the humans blood is more visible/pumping quicker than usual.
 Black eyes
When blood has not been consumed in two weeks, a vampires eyes turn black. They go from being their natural colour (red or gold depending on their diet), and darken as they get thirstier and when they finally reach 2 weeks, a vampires eyes become a completely black colour.
When a vampire becomes extremely thirsty, they will do just about anything to get blood of any type, and living in a humanised environment, this can be extremely dangerous and exposure can be risked.