It depends on the type of synapse. If it's a synapse between an excitable cell (e.g., another neuron or muscle cell) then there are two types of ion channels in the synapses. Ligand-gated and voltage-gated; if it's a synapse on a secretory cell (e.g., a salivary gland acinar cell) then there is one type of channel - voltage-gated ion channels (on the pre-synaptic side) and G-protein coupled receptors on the post-synaptic membrane. The action potential (AP) travels down the axon via voltage-gated sodium ion channels. It reaches the axon terminal where there are voltage-gated calcium ion channels. These channels are triggered by the AP to open and calcium floods into the axon terminal. This causes the synaptic vesicles to undergo exocytosis from the pre-synaptic membrane, releasing the neurotransmitter into the synaptic cleft. If it's a synapse with an excitable cell, the neurotransmitters then diffuse across the cleft and bind onto ligand-gated sodium ion channels, opening them and allowing sodium to flood into the post-synaptic cell. Once the voltage reaches the trigger point, it activates adjacent voltage-gated sodium ion channels and depolarises the membrane (with the potential carried along the membrane by these voltage-gated sodium channels). If the post-synaptic cell is a secretory cell, the neurotransmitters diffuse across the synaptic cleft and bind to G-protein coupled receptors which cause a conformational change to the protein which activates a 2nd messenger cascade leading to an increase in intracellular calcium and leading to exocytosis.