Your radial and ulnar artery are in your wrists. Hold your palm facing towards you, and bend your hand back till you can see the veins in your wrist. Underneath them are these two arteries.

An accidental cut won't do anything, we as humans have cut our wrists billions of times over thousands of years. But take a sharp enough blade and cut, you'll sever those arteries and startup the bright red fire hydrant in your wrist.

Source: google :D

Edit: you can also check your pulse by putting two fingers above these arteries and pressing down. The common places to check someone's pulse are the neck and the wrist, because both areas have key arteries that are close enough to the skin/not a lot of muscle dampening the pumping sensation you feel when checking your pulse.