The test for babinsky involves scraping a sharp object (e.g. like a pen or a car key) along the underside of the foot. In most people there is either no reaction or your toes bend forward/downward a little. With the babinsky reflex your toes bend backwards/upwards instead. Just after my daughter was born a few months ago I tried it just with my finger along her foot and it went backwards.

The theory behind these reflex tests is interesting - basically in your body there are a lot of primitive reflex arcs where a nerve detects a stimulus it is sent to the spinal cord and that links directly to a motor nerve which fires straight away and moves a muscle (without going via the brain). As our nervous system matures, our brain gets involved in modifying these reflex arcs by sending dampening/control signals down the cord to modify the reflex. If you have damage to the cord between your brain and the level at which the reflex arc nerve roots enter and exit, then the modifying signal from the brain doesn't get through properly, and so the reflexes revert back to their 'primitive' state. This is why cervical spinal cord damage can result in 'hyper-reflexivity' and things like the babinsky reflex - because there is no modifier signal, the reflex just pops at full speed without tempering/modification from the brain signal getting involved.

(note - this is only my laymans interpretation of what I've read and could be a little wide of reality ;) )