An espresso machine is not always a complex device. More sophisticated home machines have a bunch of moving parts and are therefore more prone to malfunction, although they will make better coffee when they do work. Here are some solutions to common problems.
It's not getting hot in here
First of all, if your machine isn't heating up at all (and it's plugged in, turned on, etc.) there isn't much that can be done without opening it up and tinkering around inside. Whether you do this yourself or take your machine to someone who actually knows what the hell they're doing is up to you.
No water comes through
If the machine just sits there choking on air, you may have to prime the pump first. This is typically done by opening the steam valve (on the steam wand) and letting it run for half a minute or so. Obviously, this won't affect pumpless machines or machines with self-priming pumps.
This may seem obvious, but if your machine has a removable water reservoir, check the valve on it. Then check if the reservoir itself is seated properly in the machine.
It's possible the machine is clogged. In my experience, the group head (the shower head) can clog easily. You may want to remove this — consult your manual, but it's usually removed with a single screw. If it looks clogged, try running the pump without the group head. If water still isn't coming out, you may have a more serious problem that I can't help you with. If the machine runs without the group head in place, simply soak the group head in hot water for half an hour or so and then wipe the loosened dirt off with a soft cotton rag.
Smaller, but still problematic problems
The crema is the layer on top of an espresso that is light in color and has a bubbly, almost milky texture. The presence of this is a gage as to how good a machine is (to some extent.) No machine can produce a crema with burnt coffee beans or a grind that is too coarse or too fine, so make sure you're using good beans and an espresso grind when you test your machine. Also, steam-based machines won't produce much of a crema, if any.
If the water gets too hot — or not hot enough — no crema will be produced. Check the water with a thermometer when it comes out. Ideally, it should be around 195 degrees Fahrenheit.
Also, there could be a lack of pressure in the machine. Brew a single shot and time it. If it takes the machine less than 10 or more than 30 seconds from the time it starts pouring to the time the shot is done, it may be time to invest in a better machine or getting it fixed. Of course, before doing anything drastic you should try the basic cleaning procedures I've described on another page.