Well, neurons are cells in the body that send electrical impulses down the actual neuron until it reaches the end of the cell. The next neuron is waiting for it's signal, but the synapse is the connection between the two.
When this electrical signal gets to the end of the neuron, this synapse releases chemical messengers, which float across this small space of the synapse, and connects to the receptors on the next neuron. This neuron then starts to "fire" and send its electrical impulse down the cell, and so forth.
An analogy could be something like this:
The neuron is an email - it is sent, and electrically travels through the "wires" until it reaches the end (a computer). Now, to send the message along, it has to be printed out (the neurotransmitter release) and handed to the next computer (the space between the two computers is the synapse). When this printed email arrives at the next computer, it is typed in and "sent" to the next computer, where it has to be printed out and carried to the next one, and so forth.
In this way, the electrical signal with the correct message eventually gets to the final place (which can be the brain, or an organ or a muscle) and the basic message (of the email, e.g. contract that muscle) stays the same all the way through.
Here's a much more scientific explaination: