How does a chemical synapse transmit an action potential?

The action potential reaches the pre-synaptic knob causing depolarisation. This causes voltage-gated calcium channels in the pre-synaptic membrane to open. Calcium ions enter the pre-synaptic knob through the channels along their electro-chemical gradient. The calcium ions cause synaptic vesicles filled with a chemical neurotransmitter to move towards and fuse with the pre-synaptic membrane. The neurotransmitter is released into the synaptic cleft by exocytosis. It diffuses across the cleft and binds to receptors on ligand-gated sodium channels on the post-synaptic knob. The binding of the neurotransmitter causes the sodium channels to open. The influx of positive sodium ions through the channels causes a depolarisation in the post-synaptic neurone and an action potential is generated!
(The neurotransmitter is broken down by an enzyme and removed from the cleft so it does not trigger a second action potential.)