An inhibitory synapse works just like an excitatory one!
When a presynaptic neuron fires it will release a neurotransmitter at its terminal(s). This neurotransmitter can be excitatory or inhibitory, the main excitatory one in the central nervous system being glutamate and the main inhibitory one GABA.*
GABA and Glu are far from being the only neurotransmitters in the brain, they're just a classic example, so we'll stick with them. When the neurotransmitter is released it binds to receptors on the postsynaptic neuron (provided, of course, that the postsynaptic neuron expresses these receptors).
Various GABA and Glu receptors exist, both ionotropic (i.e. channel-receptors that let ions flow through the membrane upon binding of the ligand) and metabotropic (i.e. receptors which activate an intracellular pathway that does not per se start the flow of ions, but that can induce it or prevent it indirectly).
For simplicity we'll stick to ionotropic receptors.
Glu binds to three types of ionotropic receptors: AMPA, NMDA and kainate receptors. These have different kinetics/properties, but the bottom line is that they let cations (positively charged ions, such as Na+ and Ca++) into the cell. When this happens a postsynaptic depolarization happens, which is named EPSP (excitatory post-synaptic potential).
So, if the resting membrane potential was, say, -57mV, it will become, for instance -52mV.
This means that, if the threshold potential for firing an action potential were -43mV the cell, which first needed a 14mV depolarization to fire now will need a 9mV depolarization.
If subsequent EPSP sum they can depolarize the cell sufficiently to reach threshold and let the cell fire.
This image from Wikipedia is quite self-explicatory: in this case 3 synaptic events generated 3 EPSPs that summed, making the cell depolarize enough to reach threshold potential, and generate an action potential, that will then propagate to the cell body.
GABA, on the other hand, binds to the GABA-A receptor, which is a chloride channel. In most cases, upon binding of GABA, GABA-A lets Cl- in the cell, effectively hypopolarizing it and generating an IPSP (inhibitory post-synaptic potential). The situation is the same (but opposite) to Glu, this time, though, the potential becomes more negative.
EPSP and IPSP can and do happen at the same time: as they can vary in frequency and intensity depending on the firing frequency and firing pattern of the presynaptic neuron, a pretty much continuous range of voltages can be achieved in the postsynaptic neuron.
Other controls over this process come from metabotropic receptors that can, for instance [de]phosphorylate (add or remove a phosphate group) ion channels modulating their permeability to ions or from the different kinetics of the different channels (for instance certain channels stay open for longer or open in a delayed manner etc), allowing for fine-tuning of the system.
*I am making a gross generalization here. A neurotransmitter is not excitatory or inhibitory per se, it depends on the context. For instance excitatory GABA synapses do exist.