Microsoft announced a while back that there was an electrical problem with some of the
early version (pre October 2003) Xboxes, and they would replace your Xbox power cord for
free if your Xbox was manufactured during certain time periods. Microsoft says that there's
a 1 in 10,000 chance that you'll have a problem. As far as I can tell the only damage done
was some scorched carpet, nobody's house burnt down so there's no need to panic. It
appears that the offending part is the Foxlink power supply unit (PSU) that came with
version 1.0 and 1.1 Xboxes. Getting a new power cord won't make a lick of difference to a
damaged PSU but getting the free safety cord is a good idea no matter what version you
have. They have a safety switch that kills the power to the console if it senses something's
wrong, just like they have on blow dryers. Call 1-800-4MY-XBOX and request a new cord.
They'll ask for your serial number and manufacture date shown on the sticker on the bottom
of the console. If this whole power supply thing is a real concern for you, tell them so.
They've been very helpful so far and hopefully will keep it up for a while longer.

Meanwhile, the real problem is in the power supply unit itself, behind the socket where you
plug your power cord into your Xbox. The socket's connection to the rest of the power supply
unit can come loose over time, usually from being used frequently or if the cord has been
pushed up against a wall. What happens is that you'll hear sparking or there will be an
electrical burning smell when the electricity passes through these loose connections. This
can go on for quite some time before the Xbox stops functioning altogether and you get no
signs of life whatsoever when you plug it in. These connections can be re-soldered if you're
good with that sort of thing. If you're not that way inclined just replacing the whole thing is a
lot easier and guaranteed to work.
No lights, no sounds, nothing. This isn't something that comes on gradually though, just all of
a sudden poof, the Xbox is completely dead. If your Xbox is the victim of a power surge or
nearby lightening strike sometimes a new PSU is all that's needed. Keep in mind though that
if the event completely took out every electrical appliance in the house your Xbox
motherboard could well have suffered damage and a new PSU won't fix it. If only the Xbox
and your favorite alarm clock died chances are it's just the PSU.

If you're having any sort of power supply problem the first thing you should do is borrow a
friend's power cord and make sure your existing power cord isn't the culprit. If it is, just buy
a new one and you're good to go. If you have an older Xbox you might even be able to get a
freebie from Microsoft :-) Also look in the XBox power cord socket and make sure there's
nothing stopping the cord from plugging in properly.  Other than that, fixing or replacing the
PSU will almost always solve the problem.

There are many cheap "fits all versions" PSU's available for about $20. They have a huge
failure rate and do not in fact fit all versions. I tried some in hopes of keeping costs down
but they turned out to be a particularly good example of getting what you pay for so that
was the end of that. I understand wanting to save money though so if you do decide to go
with a cheapie just make sure wherever you buy it from will take it back if it doesn't work.

The only place I know of with top quality Xbox PSU's available is Llamma.com. They're good
folks to deal with and they sell  Minebea brand PSU's which are definitely worth the money.  
Unlike the DVD drives, different version Xboxes need different PSU's. If you don't know
what version you have you can go to your favorite search engine (I use Yahoo) and type in
the words - xbox linux versions - and look for the one in English with HOWTO in the URL.  It
takes some figuring but that's the most reliable version checker I've found as the ones that
just go by manufacture date are often wrong.