What is the relationship between the structure and function of arteries, capillaries, and veins?
In short - Arteries are strong, thick vessels that can carry blood at high pressure. Veins are weaker vessels that carry blood at lower pressure and have valves in them. Capillaries are very small and have a high surface area to volume ratio.
An arteries job is to carry blood (at 'high' pressure) from the heart to the rest of the body. As the pressure of the blood is high, then the artery needs to have a thick wall that does not 'stretch' or flex according to the pressure (if the artery wall stretched, then that would ultimately lead to a decrease in blood pressure).
A veins job is to carry blood (at 'lower' pressure) from various parts of the body back to the heart. As the pressure is low, to prevent blood from 'running back down' the veins, valves are used. The valves effectively only allow the blood to travel in one direction - towards the heart. Arteries do not need valves as the high pressure of the blood from the heart automatically stops any blood from 'running backwards'.
This also means veins do not need to have such a thick wall, as they do not need to withstand higher blood pressure and thus are less likely to 'change' shape.
The role of capillaries is to get blood to the areas required in the body for 'exchange'. They can be considered the 'back roads' of the cardiovascular system (if you imagine the arteries and veins as motorways carrying lots of blood, then smaller roads are needed to 'branch' of from the bigger roads to get to where the blood needs to be).
Because of this, they are very narrow have very thin walls (to allow fast and easy transfer of substances from the blood to nearby cells). Capillaries try to have as large as surface area to volume as possible and so are very small and delicate. This is humans bruise very easily - capillaries are easily damaged and thus blood can escape into neighbouring areas unintentionally.