The level an impulse must exceed to cause a neuron to fire.

For example:
When you simply touch a doorbell, it does not ring. You must exert a certain amount of force to cause the desired reaction. It is the same with a neuron; the threshhold of excitation must be met before it can fire.

A neuron firing is simply when a neural impulse (action potential) travels down an axon, kind of like a fuse burning from one end to the other. This process is repeated innumerable times across innumerable neurons before a message can reach its destination.