Angel's curse is a very interesting and important part of the shows. Over the years, the details around it have changed somewhat. First off I'll address what exactly the curse does, then what cures it.

The curse is designed as a punishment. The vampire's human soul is restored, pushing the demon one into submission, but not getting rid of it completely. In the first season, the curse seemed to be referred to as giving a conscience to Angel, not really restoring his human soul or making him what he was before a vampire. In the second and third seasons, the curse seemed to be thought of as restoring his original human soul, making him pretty much what he was before a vampire, just in a vampire's body. When Angel got his own series, the curse is shown as a mixture between the two, more so using the first season's idea. Of course, you have to wonder what a soul is, does it define who you are? or does it just give you the urge too do what's right. I'm going to go with the second idea here. It's the memories which make a person who they are. However the soul would drastically change how you interprtate your memories. A human soul would make you feel good about what you did right. A demon soul would make you proud of what you did evil. This can make the personalities vary dramatically. So basically, a vampire is pretty much a person whose soul has changed, instead of having the urge to do what's right, you have the urge to do evil. This way, Angel is still the person he was as a vampire, and the person he was as human.

Now, what can cure the curse, and restore Angel's demon soul has also changed over the seasons. In seasons two and three, "the one moment of happiness" referred to sex alone. In the Angel series it's said that any moment of pure happiness can do it, not necessarily all sex can do it. The reasons of its changing is understandable, Angel turned when he had sex, and so logically everyone concluded that is what would change him. After thinking about it more, the characters decided that it's better safe then sorry, and all forms of happiness might do the trick. Up until recently it was still debatable as to if all sex could change him. However, Angel had sex with Darla and it did not turn him (although he did experiance the pain that happened just before turning.) The only other thing that has turned Angel was, Doximall, a powerful tranquilizer. It made him high, inducing pure happiness, proving that not only sex can break the curse. However, because it was synthetic happiness, the effects were temporary. It's funny how the curse worked that way, one would think because the happiness was synthetic, that they would not work at all, rather then just being temporary. This must just be a sort of 'bug' in the spell.

Tools needed in the spell to restore Angel's soul