Arteries, veins and capillaries are all types of blood vessels. The best way to think about their differences are the roles they play in circulation. So, if we take the systemic circulation as an example (that is the circulation from the heart to the body and back to the heart) we can look at the key features of each vessel.
Arteries are taking blood away from the heart (this is oxygenated blood in the systemic circulation). So, they have to deal with the big pump of high pressure the heart generates with each beat. Therefore, they need to have:
Thick muscular walls
Plenty of elastic tissue to spring back to their original shape
A thin lumen to keep pressure high enough for it to reach the tissues
Next the blood flows into capillaries. These are where the blood deposits oxygen into the tissues so they can respire. So, if you think about how gas transport works they need to have:
Very thin walls - one cell thick
There needs to be lots of capillaries to supply all the tissues properly
So there is low pressure because the blood has spread from few arteries to many capillaries
Finally, the deoxygenated blood flows into veins to begin its journey back to the heart. Now remember the capillaries contain low pressure blood to pass on to veins so the veins don't need thick walls like arteries. But without high pressure to keep the blood moving they will need a way to keep blood moving in the right direction (often against gravity - i.e. from the leg).
Thin walls
Valves to stop blood going backwards
Large lumen
So that is the key differences between the structure of these vessels varies and how it relates to their function.