When the action potential arrives at the axon terminal it ends; the signal. though, is conveyed to the next cell via the release of a chemical messenger or neurotransmitter.

Exocytosis: Within the axon terminal are vesicles, small "sacks" made of membrane, in which are located thousands of molecules of neurotransmitter.  The arrival of the action potential triggers the process known as exocytosis: many of the vesicles move to the presynaptic membrane, fuse with it, then dump their contents.

Receptor-binding: The neurotransmitter crosses the gap between the cells and binds with specialized sites on the postsynaptic membrane.  These receptors control ion channels; the binding of the neurotransmitter causes the receptor to change the permeability of the postsynaptic cell to some ion.  With the change in ion movement across the membrane that follows, the postsynaptic membrane changes its potential.  The neurotransmitter then is either reuptaken into the presynaptic membrane for re-use, or is chemically inactivated.

Postsynaptic potentials: The postsynaptic potential caused by the binding of neurotransmitter and receptor and the ensuing ion movement travels by cable properties throughout the postsynaptic cell, getting smaller as it goes.  Some small remnant of that PSP reaches the axon hillock, the region of the neuron where the threshold is typically smallest.  A single PSP will rarely cause the neuron to fire, but remember that the neuron might have as many as 10,000 axon terminals making contact with it, and any number of them might be active at a given time.  All PSPs occurring within the neuron at any point in time can add up, or summate, at the axon hillock.  If the result of the summation is a large enough depolarization (loss of potential -- movement from the resting potential of -70 mV toward 0) threshold will be reached and an action potential will occur.

EPSPs and IPSPs: Depending on the ion movement triggered when the neurotransmitter and receptor bind, a PSP might be excitatory (EPSP) or inhibitory (IPSP).  An EPSP moves the neuron's potential toward its threshold, making it more likely to fire; an IPSP moves the potential away from threshold, making the cell less likely to fire.  The task of the soma or axon hillock is to integrate all of the inputs, adding up EPSPs and subtracting IPSPs, in order to determine whether or not to fire.  This is happening constantly at all 100 billion neurons of your nervous system.