The Three were a group of warrior vampires who The Master sent to kill The Slayer. They almost succeeded, but Angel was able to help her get away from them and hide in the safety of her house. The Three offered their own lives to The Master because they failed in their duty. The Master accepted their offer, and had Darla kill them.
The Three were fairly cool guys. It's not often you see a group of vampires so strong and well organized that Buffy and Angel can't take them down in a simple fight. They were obviously very old, probably 400 years or more. This can be told by their strength, their code of honor, and perhaps that nifty armor they had. It doesn't seem that they were actually part of The Order of Aurelius, but they obviously respected it. It's possible they were once part of it, or it's just as easily possible that they are respectful of all vampires who, like them, follow the old ways. It's unsure what payment, if any, they were going to get for killing The Slayer. Itís even possible that they were doing the job for free. Ancient vampire honor can be quite confusing.