So what is an action potential? The action potential is is a redistribution of electrical charge across the membrane of the neuron. This redistribution occurs by the movement of ions - sodium ions moving into the cell and potassium ions moving out. In its resting state the inside of the neuron is negatively charged with respect to its outside environment. When an action potential occurs the situation reverses and for about 2 milliseconds the inside of the neuron becomes positively charged with respect to its outside environment. This signal is the basis of communication throughout the entire nervous system.
How does the action potential allow neurons to communicate with each other? This is where synaptic transmission comes in. In the nervous system the sending end of a neuron, called an axon terminal, is in close proximity with the recieving end, or a dendrite of another neuron (note that axons and dendrites aren't in actual physical contact with eachother!). When an action potential occurs a whole lot of complicated chemical reactions occur in the axon terminal which ultimately leads to the release of a neurotranmitter into the space between the axon and dendrite called the synaptic cleft. A neurotransmitter is a chemical messenger in the nervous system which allows communication to occur between many different neurons. The neurotransmitter then binds to a receptor on the dendrite (similar to a key fitting into a lock). This causes chemical changes to occur in the second neuron that could lead to an action potential.
This is a pretty simple outline of what happens because it gets a lot more complicated than this. Basically neurotransmitters are responsible for a lot of what we do. They give us control over our muscles and allow us to walk, run and dance and they can cause us to fall in love or feel sad. An excess of one neurotransmitter can lead to a disease such as schizophrenia while too little of the same neurotransmitter results in Parkinson's disease. Studying this can get to be a whole complicated mess, and the idea that some chemicals floating around in your body make you do everything you do can be a bit mind boggling at times, but it sure is amazing how each and everyone of us can manage to make and send neurotransmitters without even thinking about it.