What is neurotransmission and how does a chemical synapses work?

Neurotransmission is the process that uses electrical or chemical signals to communicate between neurones, leading to the co-ordination of secretion, muscle contraction and organ and brain function in the body. Neurotransmission takes place at structures called synapses. 
Chemical Synapses transmit information using neurotransmitters - Acetylcholine (ACh) is an example of a Neurotransmitters. Neurotransmitters are stored in synaptic vesicles in the presynaptic neurone.  Transmission across a synapse:
The arrival of an action potential at the end of the presynaptic neurone causes calcium ions protein channels to open and calcium ions (Ca2+) to enter the synaptic knob by facilitated diffusions
The influx on calcium ions causes synaptic vesicles to fuse with the presynaptic membrane, releasing ACh into the synoptic cleft by exocytosis 
ACh diffuses across the synaptic cleft very quickly (as short diffusion pathway) 
ACh then binds to complementary receptor sites on sodium ion protein channels on the membrane of the postsynaptic neurone 
This causes the sodium ion protein channels to open, allowing sodium ions (Na+) to diffuse rapidly into the postsynaptic neurone, depolarising the membrane. 
The influx of sodium ions can generate a new action potential if threshold is reached ~> postsynaptic impulse will be generated 
ACh is broken down (hydrolysed) by the enzyme Acetylcholinesterase into choline and ethanoic acid, which diffuse back into the presynaptic neurone (recycling). This means the effect of ACh on opening sodium channel is short lived and so leads to discrete transfer of information across synapse (postsynaptic membrane not overestimated) 
ATP released by mitochondria is used to combine choline and ethanoic acid back into ACh, which is stored in synaptic vesicles for future use.