Vampires aren't the only creatures in the Buffyverse. It's full with many types of different demons, widely ranging in behavior and appearance.
To understand demons, we have to somewhat understand how the buffyverse works. It works from the fact that there are many different dimensions. These dimensions are simply other worlds they have no necessary similarities to our world. They are NOT "alternate earths" or "parallel universes" they are just different places.
The shows are chocked full of references to other dimensions. Angel was sent to a hell dimension in Becoming, Buffy was sent to another one in Anne. In She a woman transported other women from her dimension to the earth one to try and free them from slavery. In the last three episodes of Angel's second season, everyone went to the Pylea dimension. In The Gift all dimensions started to merge with each other, creating some freaky results. There are many more as well.
In most that we've seen, in almost every dimension, the demons are the dominant forces, with humans being slaves or not existing at all. The term "hell" comes from these dimensions. The earth dimension started off exactly the same way. Demons lived in it and it was their world. Then something happened, somehow humans took over. Demons either went into hiding or went to another dimension.
Many times on the show we have seen example of demons wanting to merge the earth dimension with another demon dimension, or open some sort of gateway between the two worlds in an effort to make earth like it once was. Lots of demons and the more traditional vampires are resentful that the earth was taken over by humans, and want to change it back. The Hellmouth is one such potential portal, The Master tried to open it. Acathla was another way of opening a portal, which Angelus tried to do. There are many ways to merge dimensions, but they're all incredibly difficult to do, which is why most demons never get the chance. Making portals to other dimensions is a little easier, but it's still difficult enough to stop most demons from having the ability.
How this whole dimension thing first started is a big question, perhaps a long time ago there was only one dimension that split into many. But I'm not going to get into this. No one knows how the real world was formed and so I'm not going to start guessing how a fictional one was.
It's probably safe to assume that most demons we see on the shows are native to our dimension. Some are dimension travelers though, and are native to other worlds. For example, Lorn, from Angel, is native to the Pylea dimension and came to the earth dimension by a portal. It's also safe to say that certain demons are native to many different dimensions. After all, humans have been shown to be native to more than one. Of course, since we don't know the origin of dimensions it's really hard to say how everyone got where.
If creatures go to another dimension after death is a big question. When a vampire is killed, is its soul sent back to the hell dimension it's originally from? My guess is no. When something dies it is gone. Darla would confirm this, as after she was resurrected, her last memories were of when Angel killed her. Not of time spent in any hell dimension.
However, this doesn't mean everyone is gone after they day, I think that maybe The Powers That Be send people to a heaven dimension after they die. Which people do they send? it's hard to say. We know Buffy went to a heaven dimension after she died. But would Angel? do all humans? do all humans go even if they're evil? do only heroes go like Buffy and Angel? Then what about someone like Giles or Xander? Maybe only "tools" of The Powers That Be go, like Slayers? There are just too many questions on this subject and no answers.
Anyways, lets get back to vampires and how they fit into all this:
At the time when humans started taking over the earth dimension, one demon bit a human, drank it's blood, then made it drink the demon's blood, which infected it with a demon soul. The result? Vampires. This demon, the Vantal, has never been seen except for when Angel was changed to one in the Pylea dimension.
Now, vampires aren't the only demons who are demon/human hybrids. Lister and Brachen demons are also both thought of as not being of pure demon blood. There are probably many others as well. An interesting thing to note is that Doyle can change his appearance from human to demon much in the same way Angel can... Perhaps many demons with human blood in them can change like this. Though evidently, not all can, as Lister demons were unable to change into a human form. I can't recall any demons confirmed as full bloodied who can do this except through magic.
By reading this I hope you now understand a little bit about how vampires fit in with other species of demons... and where they all come from.