First you must understand that the inside of the membrane is negatively charged. Outside is positively charged. Therefore, there is a voltage difference.
a neurotransmitter binds to ion channels at the synapse.. which allows Na+ in ... as Na+ rushes in... there are channels next to where the Na+ rushes in that recognize that increase in positive charge (these are called voltage gated ion channels) and open up themselves (causing depolarization)... which results in more Na+ rushing in. You get a chain reaction all the way down to the end of the neuron.
When it reaches the end of the neuron, there are voltage gated ion channels that only allow Calcium in. When calcium rushes in, it allows that neurons own neurotransmitter stores out which bind to the next neurons ion channels and the process starts all over.
That's it in a nutshell, but you need to know things like what hyperpolarization is (caused by voltage gated K+ channels)
google is your friend