The idea that neurotransmission occurs at synapses and is mediated by chemicals was, at first, a contentious issue.[1] It was in the first half of the 1900s that experiments proved chemical neurotransmission occurred.[1] 

The process is outlined in brief on the slide.[1,2,3,4] Briefly, an action potential arrives at the terminal end of the presynaptic neurone, which means that the membrane potential at the terminal is altered.[1] This altered potential causes voltage-gated ion channels to open, leading to a rapid influx of calcium (detail not shown on slide).[1] The increased calcium within the neurone terminal (which is just a transient effect) causes vesicles to fuse with the membrane, and release their contents into the synaptic cleft.[1] Among the released contents are neurotransmitters, which bind to specific receptors on the postsynaptic neurone.[1] The effect of this binding varies, but usually leads to altered conductance of the postsynaptic neurone, and the propagation of the action potential.[1]

Vesicle – a membrane-enclosed sac that stores or transports substances.
Receptor – a protein molecule on a cell membrane that binds to a specific chemical, such as a neurotransmitter or drug, and produces a specific physiological effect.  Reuptake – a mechanism by which a neurotransmitter is taken back into the axon terminal that released it; the most common mechanism for removal and inactivation of neurotransmitters.
Astrocyte – a class of glial cells; functions include maintenance of extracellular ionic environment, and structural and metabolic support of neurones.
Diffusion – in regard to neurotransmission, a mechanism by which neurotransmitters drift out of the synaptic cleft.

References:
[1] Purves D, Augustine GJ, Fitzpatrick D, et al. (eds). Neuroscience. 4th edition. Sinauer Associates ,2008.

[2] Tortora GJ, Derrickson B. Principles of Anatomy and Physiology. 12th edition. John Wiley & Sons, 2009.

[3] Kandel ER, Schwartz JH, Jessell TM (eds). Principles of Neural Science. 4th edition. McGraw-Hill, 2000.

[4] Sadock BJ, Sadock VA, Ruiz P (eds). Kaplan & Sadock’s Comprehensive Textbook of Psychiatry. 9th edition. Vol 1–2. © Lippincott Williams & Wilkins, 2009.