The stories are primarily about "Dragonlords," weredragons that can change from human to dragon shape. Originally there were just dragons and humans in this world (well, not just, there are several other sentient and partially sentient species, but those are the two that matter in this particular case.)
A couple tens(?) of thousands of years ago there was some big magical war between the various tribal human groups around at the time. At some point there was an apparently self-propagating magical accident that resulted in a human soul and a dragon soul getting merged together. However you've now got two souls taking up the "space" of one soul, which i guess is unstable or something. So the new soul (presumably now well mixed) splits into two souls again, and each half gets born into a weredragon. So only Dragonlords have soulmates in this world, because there is someone out there who literally has the other two halves of their own soul(s).
The Dragonlords are all originally born as "truehumans," as the Dragonlords call them. Most of them are sickly as youths, and all of them have some kind of "Marking," which is a physicaly deformity or some other kind of oddity. Some of the deformities are obvious birthmarks, being a midget, being albino, or having six fingers, although other less obvious Markings sometimes occur, just to confuse everyone. And of course not everyone who has something that _could_ be a Marking is actually a Dragonlord.
So if they survive their frequently sickly and possibly harassed youth, at some nonspecific time (the ages of 15 and 28 have been sited in two particular cases i believe) they undergo their first change and become a Dragonlord. They spend the next many(?) millennia as Dragonlords (the only specific benchmarks i remember is that 600 or 1000 years is still considered young) until they get tired of life. At that point the human half decides to give up and fade away, at which point the dragon half (which is usually mostly dormant up until that point) takes over and they become a truedragon.
All the Dragonlords live at Dragonskeep, where they spend most of their time when they're not out adjudicating disputes among the humans. (Part of the aftermath of the war that formed the Dragonlords was a pledge that the Dragonlords wouldn't make war on the humans or attempt to rule them (i think) but they would act as final arbitrators when requested.) So after a Dragonlord undergoes their first change they go to Dragonskeep and either find their soultwin, or find that their soultwin isn't around yet and they need to wait for possibly hundreds or thousands of years.
Unlike a lot of books with soulmates in them it isn't made out to be 100% positive in these books. Most soultwins feel close to each other, some incredibly close, others about the same as "normal" human lovers. However some smallish percentage of soultwins can not stand each other and do their best to stay away from each other as much as possible, presumably beause one or both of the souls has psychological issues. Also, when the souls are joined there's a small risk that the two (four?) halves will decide that they don't want to be apart again, in which case one or both (not clearly stated) of the Dragonlords dies when they don't get their soul back. And of course there's the usual what happens when your soulmate dies. In this case the human half usually gives up on life and commits "suicide" by handing over the body to the dragon soul.
It's not explained what happens to those Dragonlords whose soultwin dies before their first change. My guess is that they don't become dependent on their soultwin until they're joined for the first time, prior to that they just don't know what they're missing perhaps? Or maybe it's similar to a drug dependency. In any case they probably jsut continue to be lonely and occasionally depressed like other unmatched Dragonlords.
As for the books themselves, as i've already said above, i enjoy them. She has a somewhat disjointed style, jumping from one scene to another and back again. That's certainly not unusual, at least not in the books i normally read, but she seems to switch much quicker than most other authors do. Sometimes she'll stick to one point of view for a fair bit of time, but sometimes she'll spend less than a single page on a scene before switching to something else, Sometimes she'll cut away right in the middle of an action scene and then cut right back into the same action scene a few paragraphs later. This can be confusing and hard to handle at the begining, and almost put me off "Dragon and Phoenix" at more than one point. However once you get past the first 100 pages or so and you get to know the various threads it kind of sucks you in.