The first vampire was made when a demon fed off a human, and then mixed his blood into the human. Nothing has really changed.
A vampire can be pretty choosy about who they make a vampire, and normally will only sire if they have a use for the person's abilities. Why more vampires don't create armies of minons is a question many people wonder. Basically, you have to think of it this way, if you were a vampire leader, you would have others wanting to take your place. Remember, vampires feel no guilt and so would not feel bad if they killed their sire. Keeping a small army is much more smart when it comes to most vampires. There are of course exceptions, The Master had no army, but he had quite a lot of minions. Most had been loyal to him for hundreds of years and would protect him no matter what. Because of this, The Master could have more minons working for him. He was also incredibly powerfull himself, so a takeover would be almost impossible.
The rules on siring are not totally clear. Basically what happens is a vampire feeds normally, and then the person being fed on will start feeding back on the vampire. It would seem that a vampire can not drain all the human's blood, or they will be unable to be sired. It's unclear as to why the other person will feed back on the vampire. One reason could be that they give some kind of telepathic or hypnotic suggestion, extremely powerful vampires like The Master have been shown to be able to do this in normal circumstances. Maybe all vampires have this ability, but can only use it when a person is severely weakened, and even then it is hard work (another reason why vampires don't sire more often.) After the person drinks some of vampires blood, they basically die and then after a varible amount of time are reborn as a vampire. They no longer have their human soul, meaning they will lack their values of not doing wrong, and will not feel guilt over doing wrong. They also have what the show refers to sometimes as a demon soul, which makes vampires have a love for carnage.
Let's backtrack about to the dying, or dormant period of a siring, the person will lay dormant between a few hours and a few days. There is no really evidence as to why there's such a huge gap except for the fact they wake up when the plot needs them too. Obviously, the reason the time varies is because it's need for the story... but this site is about making sense of everything on the show, so I'll give it a crack. It is my opinion that the vampire's demon soul does not take over until it senses that it is safe or ment to do so. This would explain why certain vampires wake up when they do, and why they would not wake up in the middle of the day. It is also my belive that the sire can give the new vampire a subconscious suggestion before they rise. This message could be when to wake up. For example, the vampire Vorba woke up in a funeral home because he was ment as a decoy. Perhaps his sire planted the idea that he had to get up when the Slayer came looking. Another example would be when a woman Angeles sired, woke up in the middle of a funeral home when the Slayer came to look for her. The woman, who was Buffy's friend when alive, told Buffy a message from Angel. Perhaps that was planted too?
The whole soul thing is kind of confusing, so I'll go into that more. From what I can tell, the only thing a human soul does is make people have the urge to do good, and makes a human feel guilty if they don't. A soul is not personal, meaning a soul doesn't make a person's personality, if Buffy has Willow's soul Buffy would still act exactly the same. This is why vampires will often retain personality traits. They have the same memories of the human, and therefore have had the same experiences. Is a person really no more then the sum of their memories? The reason that the personality can differ so much is because unlike a human soul, a demon soul has no conscience, no remorse, everything the human had thought about doing, they could now do it with no regret. They also will have the urge to kill and be evil. This would drastically change how a person behaves. Angel has both of these souls in him but he keeps in demon soul burried. I go into souls a lot more in many other of my pages, so be sure to have a look there.