How is a message transferred across a synapse?

A synapse is a junction between the end of one neuron (known as the pre-synaptic neuron) and either another neuron (known as the post-synaptic neuron) or another tissue such as muscle (such as at the neuromuscular junction - the clue is in the name!). Either way, electrical signals are being transferred to chemical signals and then transferred back to electrical signals so that the original message can be passed on.
The word synapse ecompasses everything going on to transfer the signal at this junction. The synaptic cleft is the space in between the pre-synaptic terminal and the post-synaptic terminal. This space is tiny, only about 10-50nm (over 100 times smaller than a millimetre).
So how does it work?
The electrical signal - an action potential - arrives at pre-synaptic axon terminal. This opens channels that are known as voltage gated calcium channels. They are voltage gated becuase they respond to changes in voltage at the nerve terminal and they are calcium channels because only calcium ions can flow through them. 
When these channels open, calcium flows into the pre-synaptic terminal. Calcium then binds to a protein called synaptotagmin which detects the rise in calcium. This then leads to proteins known as the SNARE proteins to fuse. This SNARE protein complex guides vesicles of acetylcholine (a neurotransmitter) right to the end of the nerve terminal where they bind with the membrane and are released into the synaptic cleft. This is known as exocytosis. Acetylcholine diffuses across the cleft to the post synaptic terminal where it binds with receptors known as nicotinic acetylcholine receptors. The cleft is small so that the neurotransmitter reaches its target easily. When binding occurs, sodium flows through the channels, depolarising the membrane and causing muscle contraction to occur. 
When we have opertions, it is important that muscle contraction doesn't occur so drugs like tubocurarine act to block these receptors.
Botox is another example of disruption of the neuromuscular junction. Botox (actually called botulinum toxin) stops fusion of the vesicles with the membrane so exocytosis cannot occur. This paralyses the muscles and some people use this to smooth their skin or to stop twitches. 
Why have synapses at all? Why not just have direct electrical connections between nerves and muscle? Well, synapses hugely increase the diversity of signals that can be sent within the body. The best example is in the brain. An adult brain has one hundred billion neurons and each one of these has roughly 7000 synapses connecting with other neurons. Messages are sent and filtered via these synapses to allow normal functioning of the brain. New synapses are made all the time when we create memories or learn something new. Synapses are more easily made when we're young so that's why people say it's easier to learn languages when you're young. Synapses can be lost too, especially in diseases like Alzheimer's Disease which affects your memory.
I hope that is helpful. Looking at a picture of the process can make it easier to understand so check out your textbook or look up on google images.