Death and clones

If your ship gets blown up, you end up in a little pod in space. You should have purchased Insurance for your ship unless it's your Rookie ship. The pod can warp. Once back at a station, you will get a free Rookie ship. Before you engage an enemy, it's wise to have a destination set in case you need to quickly jump in your pod or ship.

You will lose a lot of ships while you learn the ropes. When you buy a new ship for yourself, insure it. Losing a ship is a normal part of the game, with just a financial cost of refitting the ship you buy to replace it. Insurance will not cover the cost of your fittings (guns, armour, boosters etc).

NPC enemy ships do not attack your pod. Players might. If you are killed in your pod, you revert to your clone. That's a real death in Eve.

Medical clones
You start the game with a clone. They are like save-points for your learned skills. They are located in a space Station. That's where you resurrect when you're pod killed. Once you are pod-killed, your clone is activated, and you must buy a new one.

Dying without a clone means you lose a lot of skillpoints. It's really bad. If you're an experienced pilot, it might be a month's full-time training or more. You lose 10% of the skillpoints on your most invested skill.

You can move which base the clone is in, to be nearer where you are adventuring. The medical service in-Station lets you do that.

Clones have a capacity. It needs to exceed the skill points you have altogether on your character. The clone is automatically and continually 'saved'; you don't do anything for it to store the skills you've learned. They just have a ceiling on how many points they protect. Your first noob clone can protect 900K skill points. You'll need to upgrade it after a week or so.

Jump clones
First off, there is generally much confusion about the mechanics of jump clones. For starters, to install a jump clone, you must have either a personal or corporate standing over 8.0 in the station you are trying to install the clone in. The station must also have medical facilities. Jump Clones require learning the 1M ISK skill 'Infomorph Psychology'.

Once a jump clone is installed in that station, think of it as a destination. The best way to understand how jump clones work is to completely separate the ideas of your medical clones and your jump clones. What happens to one does not affect the other.

For example: You install a Jump Clone in station 'Alpha', and then leave the station and fly to another station we'll call 'Bravo'. Once you get to Bravo, you can choose to jump to Alpha, regardless of your standings with the owner of Bravo station, and regardless of whether or not it has a medical facility. When you jump, you become the clone and are instantly moved to station Alpha, and your previous clone (at origin) is left behind as a new jump clone in station Bravo, thus becoming a new destination.

You can only clone jump once every 24 hours, and it requires you to pause your skill training. You will only get the effect of implants installed in your current clone. When you are pod-killed, you will always go back to your Medical clone station, and any implants in your current clone will be destroyed. Being pod-killed does not, in any way, affect your jump clones!

(Text on jump clones from Cordova, of Vendetta Underground).

Further reading
This article from Agony Unleashed talks about suicide jumping, more on implants and jump clones.