Why did my computer make a popping sound, smell like it was burning, and then die?”?

Does the machine still work?
My first guess is that you may have fried the power supply, or possibly one of your cards (or something else). Sometimes it’s just a part that’s had enough, though you’re more likely to be looking at a power surge. Sniff around the back of the case, where the plug comes in from the wall, for more burnt-plastic smells.

Most desktops can be opened with a screwdriver and some perseverance; if you’re confident (and won’t be voiding anything by doing so), crack the case and examine the components for scorch marks. Unplug the machine and touch something metal (in order to discharge any static you might be carrying) before digging into the guts.
Dell cases are especially user-friendly, not even needing a screwdriver. Tampering with them might disqualify you from any Dell-based aftermarket service.

Try not to touch anything while you’re in there.

Computers are modular by design; so long as damage isn’t sustained to the drives or the motherboard, you – or the guy at the shop – can swap out the bad bits.

@Nullo pretty much has it covered, to be more specific though, it sounds like it busted a cap.

That is, a capacitor popped. Capacitors are the things that look like little metal cylinders wrapped in plastic on motherboards, video cards, in power supplies, etc. Look (and smell) for one that where the top is cracked open. There might be goo around it. If you find it, then you know which part needs to be replaced. (and like @Nullo says, if it’s the motherboard, then replacing it is hard, but not impossible. May be easier to build/buy a new computer and take some parts from the old one though.) -

Also, if the computer Does still have a good warranty then ignore everything above and just send it in for service.

Last time I did that, it was the power supply self-destructing. I was lucky in that the PSU was the only thing that fried.

On a related note, do not get a cheap power supply! Spend the extra few bucks for a name-brand one like Antec, Cooler Master, Thermaltake, or Corsair (there are a few other good ones that aren’t at the tip[ of my tongue right now…) A cheap PSU can fry pretty quickly and may take the rest of your rig with it.

They pretty much already gave you the answer, if you do open up your computer just be careful, some of the capacitors may still hold a charge, though the power supply will lower the voltage there are still very high voltages (over 2000KV) inside your computer for the monitor, just watch where you put your hands! I recommend taking it to someone who knows what they are doing if you are not covered by the warranty.

@Jabe73 Personally, whenever I turn my computer completely off, I hit the PC’s power button after hitting the master switch on my UPS. That way, the thing is discharged.