The tiny visible vessels within the skin are normally referred to as spider veins (also known as telangiectasias).   Spider veins appear mostly on the legs or face and can be removed in two ways.

SCLEROTHERAPY INJECTIONS
The first option that you have is to have them injected with a sclerosing solution causing them to contract and collapse – this procedure is called sclerotherapy.    The spider vein will first turn white (blanche) and then disappear. Since a micro-needle is used this form of treatment is best suited for larger and medium sized spider veins.
It is a relatively painless procedure and no "down-time" is required, although you may have to wear a bandage or compression hose for 5 to 10 days - depending on the extent of work done.    Multiple treatments are typically required, and some veins may refuse to respond to sclerotherapy.

LASER VEIN REMOVAL
Treatment of spider veins with laser is best used on small to medium sized ones.       When the laser is passed over the spider veins they coagulate and shrink, and after treatment look darker and then over a period of up to six weeks will fade.

Even if you have a 100% removal rate of spider veins, new spider veins may appear in time, since these therapies are for the removal of already formed spider veins, but does not prevent them.