A synapse acts as a junction between cells - either between neurons, or between a neuron and a muscle or gland cell.
Although neurons transmit information via electrical signals, synapses transmit information rapidly via chemicals - these are called neurotransmitters.
When an action potential reaches the end of a neuron (called the pre-synaptic neuron) - called the axon terminus - the change in potential across the cell's plasma membrane stimulates the opening of voltage-gated calcium (Ca2+) channels - causing calcium to rush into the axon terminus.
This in turn triggers the fusion of synaptic vesicles, which carry the molecules of neurotransmitter, with the plasma cell membrane - this releases the neurotransmitter into the gap (called the synaptic cleft) via exocytosis (the fancy word for transporting 'stuff' such as proteins out of a cell).
The neurotransmitter molecules with then travel across the gap and bind to receptors on the plasma membrane of the post-synaptic cell. This binding then triggers the influx of ions - normally sodium (Na+) into the post-synaptic cell. If this amount of sodium reaches the threshold potential of the neuron, an action potential will be set up in this cell.