Blood vessels are, quite simply, the tubes which take blood around the body. To put it in basic terms, they go from the heart, where blood is pumped, to the parts of the body that need oxygen, and then take the blood back to the heart again.
It sounds like an unimportant role, but in fact blood vessels are extremely valuable. As well as transporting blood, and the oxygen and nutrients carried with it, the blood vessels enable the transport of water and electrolytes, and also offer support to temperature control and immunity!
This is why it's terribly useful to note that they've been designed well for their function. The typical structure is of an inner layer of endothelial cells (i.e. a single layer of cells lining the inside called the tunica intima), then a muscular layer called the tunica media, and then an outer layer called the tunica adventitia, which contains nerve cells and the blood vessels supplying the vessel itself! The tunica adventitia actually contains more connective tissue than anything else, and has fibroblasts in it, but we don't want to get into that at the moment. The presense of these layers depends on which kind of vessel you're talking about, and they will vary in thickness depending on whether you're looking at an artery, a vein or a capillary.
An artery is a blood vessel which leads away from the heart. This mean that the pressure in these vessels will be high, so the tunica media of these vessels which contains the muscle is thick. They're wonderfully elastic vessels which can cope with the heart pumping enthusiastically all day long, stretching to accommodate each swell of blood, and subsiding once again during diastole.
They do have nerves innervating them, which means if the body wants them to change (e.g. to vasoconstrict in order to alter blood pressure) then they can do. In actual fact it is the arterioles which are altered when the body wants to change its blood pressure.
Since an artery is a vessel taking blood away from the heart, it is usually full of oxygenated blood. This is true of all arteries except for the pulmonary arteries, which are taking blood away from the heart to the lungs. Since there is a far greater number of arteries carrying oxygenated blood, it's usually assumed that when you're talking about arteries, you're talking about one of these, so the blood inside will have plenty of oxygen kicking around.
Veins are vessels which take blood back to the heart. You might hear about blood pooling in the veins, and that's because the veins act as a kind of reservoir to keep hold of blood on its way back. The vessels are a bit wider than arteries, and because the walls are thin, the tube through which the blood is able to pass is even wider. The venous system (i.e. the collection of all the different veins) is a low pressure system so the blood isn't in a rush to get anywhere; although it is drawn back into the heart, ready to zoom off to another location, it holds about 70% of the blood in the body.
Most of the veins in the body contain deoxygenated blood, because it's returning to the heart having visited the organs and got rid of its oxygen. There are usually four veins which return to the heart from the lungs (the pulmonary veins) and these obviously contain oxygenated blood; apart from that the rule works.
Because veins are in a low pressure system, they don't need a thick wall like arteries do. Instead they have a thin wall which is able to change easily according to the pressure in the system, and as a result the veins can act very well as this reservoir. Only the slightest change in shape can change the volume enormously, which means if you need to store the blood here, you can.
The other thing about a low pressure system is that it's a lot more difficult for the blood to get from one end of the vessel to the other. Especially in the legs, where the blood in the veins is going against gravity, it's got quite a job to make sure it returns to the heart. To make sure that blood doesn't flow the wrong way, veins contain valves which stop the blood flowing in the wrong direction. Like valves in the heart, the venous valves allow blood to travel in the right direction, but if it tries to flow back, the cusps of the valve close to prevent it.
The other thing which veins have is the muscle pump, where the contraction of muscles beside the veins aids the flow of the blood. Although on its own the muscle pump wouldn't necessarily help that much, because of the valves, blood can only go in the right direction, and the muscle pump helps it along.
The reason that veins look the colour that they do when you can see them is that the red of the haemoglobin in the blood is filtered out. The carbon dioxide in the blood (which looks blue) and the skin and fat overlying the vessel combine to give a greeny-blue colour that you see.
Arterioles come between arteries and capillaries; venules are the equivalent for veins.
Arterioles are like small arteries, branching off arteries and leading down to the capillary bed. They're not arteries because they're not the same, main vessels - but they still have a muscular tunica media, which is used for vasoconstriction in order to increase blood pressure. The pressure in the circulatry system drops as you go through, mainly because of the constricted arterioles that prevent high pressures getting through. If there's something wrong with these vessels - e.g. they're too constricted, or they're narrowed by fatty deposits in the vessel walls - then the blood pressure will be increased.
Venules are a bit like arteriole-equivalents, as they lead from the capillary bed back into the veins. But they're quite simple things, with thinner and less-developed walls. They still have all the three layers which most blood vessels have, but they're not substantial, and venules aren't important for controlling blood pressure.
Arterioles are the last blood vessels which could be considered arterial, as they are taking blood away from the heart towards the body's capillaries. Venules are the first blood vessels which could be considered venous, as they collect blood from the capillaries and join up together to form veins.
A capillary is the tiny blood vessel that you get at the end of the blood's route to the organs. If you're trying to supply blood to muscles, for example, then the arteries will lead to arterioles, and into the capillary bed, the thinnest of the vessels which in some cases are narrower than the width of a red blood cell!! Sometimes the cells need to squidge in order to fit through.
Capillaries don't have the three 'tunica' layers that other blood vessels do; instead they're surrounded just by the single-cell layer of endothelial cells. There is a basal lamina which is made of fibrous proteins to support these cells, but the wall needs to be kept thin so that oxygen and nutrients can get out of the blood and supply the cells of the organ. The only cells around which aren't endothelial cells are pericytes, which are a bit like smooth muscles cells, but their function isn't really understood.
As the blood passes through the capillary bed, oxygen and nutrients are passed through the wall to the cells of the organ, and waste products including carbon dioxide are thrown back out into the blood. So oxygenated blood is turned into deoxygenated blood, and carried off via the veins back to the heart.