What are the smallest blood vessels called?

Capillaries, some so narrow that red blood cells must deform and pass through in single file. Their walls are a single cell thick, which is the entire point - everything the body exchanges with its blood, oxygen, carbon dioxide, nutrients, waste, crosses there and nowhere else. Arteries and veins are plumbing; capillaries are the only place the work happens.

The total length is difficult to believe and worth stating: estimates run to tens of thousands of kilometres in one adult, and their combined surface area is comparable to a tennis court. No cell in the body is more than a few cell-widths from one, because oxygen diffuses over very short distances and that limit sets how tissue can be built.

Controlling them is how the body handles temperature. Capillary beds near the skin open to dump heat and shut to conserve it, which is why you go red when hot and pale when cold. It is also the mechanism tumours exploit - a growth beyond a millimetre or two cannot survive on diffusion alone, so it releases signals that make new capillaries grow towards it, and drugs that block that process are a whole class of cancer treatment.