If You Do Get Blown Up: If your ship gets blown up in EVE Online, it's gone forever. About half of your modules and cargo will also be destroyed, the other half will be left behind in the wreck. Chances are very good however that your attacker will loot your wreck long before you can get back in anything with a cargo hold, so the stuff that wasn't destroyed is probably a write-off as well. Still, it's not so bad.
Assuming you had platinum insurance (which you should always buy for every ship you fly), you will immediately get a
payment in your wallet for the estimated market value of the ship at
the time you bought the insurance1. Thus, you are really only out the cost of the modules and the insurance plan. All this, of course, is assuming that your escape pod gets away...
When your ship blows up, you appear in space
in a pod. The pod is considered a new ship, which means that the
people who had you locked and scrambled before will have to relock
you. Pods however have much higher agility than any other ship in the
game and can warp faster than any ship can lock. So if, as soon as
your ship is about to explode, you click on a distant planet or
station or gate or something and start spamming the warp button then,
unless lag screws you, you will always get away.
In null sec your gooey bits are in a lot more danger because people
can deploy warp disruption Area of Effect bubbles, but such things are
barred from empire space (low sec and high sec).
If you do get podded, you will lose your implants and your medical clone and will wake up in whatever station your medical contract is set to. By default this will be the rookie system where you did the first tutorial mission, but you can change this by clicking the medical button on the right hand panel in any station. You will have to buy a new clone, which will get more expensive as you get more skill points. It is vitally important to always keep your clone updated. If you get podded with an out of date clone you are at risk of losing actual skill training time that can never be regained. The basic clone that you through your trial, at least. After that you are going to want to periodically check your character sheet to ensure that your total number of skill points isn't too close to the number of skill points covered by your clone (conveniently, the two numbers are displayed right next to each other).
Incidentally, you can set your medical contract to any station you have visited which has cloning facilities, no matter how far away from it you currently are. This means that you can, with a little planning, travel the length of the galaxy by self-destructing your pod. Just make sure that your clone is up to date and that you avoid expensive implants!
So, keep in mind that death in EVE can have more significant consequences than in most other MMOs. It's possible to blow a year's worth of Isk on a handful of rare items, put them in your cargohold and then have them summarily and irrevocably destroyed. Then, to top it off, you could get podded with an out of date clone and lose months of skill training.
At the same time, by taking minimal precautions and remembering the maxim "Never fly what you can't afford to lose," you can rob death of much of its sting. And this is a good thing, because the most fun (and profitable) experiences in New Eden are, almost without exception, also the most dangerous.
1. A note about insurance: The more advanced ships in the game (Tech 2 ships, Tech 3 ships, Faction Ships, etc.) can only be insured for a fraction of their cost. But it will be quite a while before those classes of ships make up much or any of your fleet.