Assuming that the machine already has water in it, you start by turning the machine on and waiting for the heater light to indicate that the heating vessel has heated the water to the ideal temperature (just below boiling).
Next, you put the ground espresso coffee into the basket and tamp it down.
You install the porta-filter by twisting it into the machine, and you place one small cup beneath each spout. You then turn the valve to the espresso position. This engages the micro-switch that starts the pump, which pressurizes the heating chamber and hot water to about 15 atmospheres (220 psi) of pressure. This forces the hot water through the ground coffee and out of the spouts. Ideally, it should take about 25 seconds for about 1.5 ounces of espresso to come out.
When the espresso has filled the cups, you put the switch back into the middle position. Most likely, you'll want to steam and froth some milk next.
To steam some milk for, say, a cafe latte, you place a container with some cold milk under the steam wand so that the wand is submerged. Then, you turn the valve to the steam position. This energizes the resistive heater, which quickly boils the water in the heating vessel and opens the valve, starting the flow of steam out of the nozzle. The pump runs intermittently to keep the heating vessel supplied with water. The steam quickly heats up the milk, and, if you hold the steam nozzle near the surface of the milk, can be used to make froth.
There are dozens of different espresso based drinks that you can make or order at an espresso bar.
For more information on espresso, espresso machines and related topics, check out the links below.