The passage of a signal across a synapse is called synaptic transmission. A link describing the process in detail is given in the related links section. A simplified explanation follows.
Within neurons, information travels via electrical impulses called action potentials. Action potentials cannot be transmitted electrically across the synapses that join most neurons. These synapses, technically called chemical synapses, can only transmit information chemically (hence their name). So for a signal to pass from one neuron (called the presynaptic neuron) to the next (called the postsynaptic neuron), it must first be converted from an action potential to a chemical signal, and then back to an action potential. When we talk about synaptic transmission, it's this conversion of signals that we're talking about.
An explanation of action potentials can be obtained elsewhere, but suffice it to say that an action potential results in a temporary change in the electrical properties of the cell membrane; this change is called depolarization. When an action potential reaches the very last bit of the presynaptic neuron (called the presynaptic terminal), the resulting depolarization causes voltage-gated calcium channels to open. When these open, calcium rapidly enters the presynaptic cell and causes the release of neurotransmitter into the space between the pre- and postsynaptic cell (this space is called the synaptic cleft). These neurotransmitters are the chemical signal that carries information from the presynaptic neuron to the postsynaptic one. In other words, an electrical impulse (the action potential) has been converted to a chemical one (the neurotransmitter).
Upon reaching the postsynaptic neuron, the neurotransmitters bind to receptors found on the postsynaptic membrane. In a typical neuron, the binding of neurotransmitter to the postsynaptic receptor triggers the opening of ion channels. The resulting flow of ions through these channels now changes the electrical properties of the postsynaptic cell, potentially leading to depolarization and the initiation of a postsynaptic action potential. Now the chemical signal (the neurotransmitter) has been converted back into an electric impulse (the postsynaptic action potential).