Veins are blood vessels that carry blood back to the heart. Deep veins are those found deep inside the body as opposed to veins closer to the surface (superficial veins). If a blood clot (thrombosis) forms in one of your deep veins, normally in your leg, this condition is called deep vein thrombosis (DVT).
Clots can cause swelling and pain and may lead to complications such as a pulmonary embolism (i.e. the initial blood clot gets transorted and deposisted in the arteries supplying the lungs causing a blockage.) Immobility for long periods, such as plane journeys, decreases the blood flow rate through your deep veins as the muscles surrounding these blood vessels are not contracting; this promotes the formation of blood clots.