My legs are in pain after running, what could it be?

Muscle, neve,blood: Some things it can be but are not limited to include muscle fatique, exercise induced compartment syndrome, peripheral vascular disease, claudication (neurologic or vascular), myopathy, some medciations can cause muscle fatigue.

Lack of blood flow!: When we run, there is increased utilization of oxygen at the level of muscles. When there is not enough circulation to the area, Lactic Acid level goes up because body resorts to surviving on a low-oxygen environment. The pain you experience is to tell you to stop running, and take a break! warn-up before running helps.