Did you ever watch Game Of Thrones? (Dont read the rest, SPOILERSSSSSS!) in the last episode, Khaleesi does not burn because she have dragon blood in it. Means she is a dragonborn. Correct me if I am mistaken...
Well, I've read the first three books and have watched season 1 at least 3 times. Here's what happened:
SPOILERSDaenarys is the heir of House Targaryen, the very same House that descended from Aegon the Conqueror. Aegon used dragons to conquer Westeros some three hundred years ago. It stands to reason that the Targaryen bloodline has some connection with dragons and can therefore control them to some degree. You could compare them to the Dragonborn if anything though even that's a stretch. The Dothraki however are a barbarian tribe with no relation to the Targaryens or dragons whatsoever. Khaleesi is a strictly Dothraki term that amounts to queen. Notice she wasn't Khaleesi before being married to Drogo, the Khal or king of the Dothraki.