Death, or Physical Exhaustion, leads to your Anima Form, or what I like to call a ‘ghost’, appearing at the nearest Anima Well. Anima Wells are scattered throughout each zone, and upon death you can choose to run back to your body to respawn, or you can use the World Roots at the Anima Well to respawn right there.
During major boss battles in dungeons you cannot return to the fight at all after death, while the fight is still on. This makes team-play and co-operation a must, as one of your party must still be alive to take down the boss, or else you’ll have to start the battle over from scratch. Each major boss must be taken down in one go, not zerged to death in bits and pieces until you wear it down to nothing with your power to res fast.
Part of TSW’s death penalty is durability loss. Each weapon and talisman has a durability, which does not reduce during regular use, and does not effect the damage output of the weapon until the weapon is "broken". At that point you will need to visit a vendor to "repair" the broken weapon and/or talisman.
But death isn’t only for corpse running. The Secret World uses death in other, unique ways. Some missions require you to die to complete an objective, and there are some sights that can only be witnessed while dead, in your Anima Form, so keep this in mind while you’re exploring the hidden corners of The Secret World.