They meld their neural system with their mates' and kinda 'scan' their DNA. Then they absorb parts from it and combine them with their own DNA to produce a child. As for mating with animals, ewww... It should be possible, technically, but ewww...
Look and see MEwiki for details. Anyways, asari do not take any DNA from other species. They just use their telepathic connection to randomize their offspring DNA. So this means all asari children are just asari and no hybrids. Dogmatix314 21:37, November 5, 2011 (UTC)
I think it was stated in a Codex entry that some asari still believe that they take aspects of their partner's genes when they breed outside of their own species. Also, the personality of the asari bartender on Illium could be taken as a nod from the developers that asari children DO share some personality traits/frames of mind with their "father". (Since that bartender was totally a krogan in an asari body. "Family, what a kick in the quads.")
^That might have more to do with Aethyta's upbringing by a Krogan than genetics. One of those two squabbing asari near the terminals you have to hack for Liara on Ilium, when the other one tells her she is acting like a jerk maybe because her father was Batarian, she responds that her father was an excellent caretaker. I have read somewhere though that those facial markings most asari have indicate the father species and that purebloods (like Liara, Morinth or the shopkeeper at Baria Frontiers) don't have them, so the father species has some genetic impact. TiberiusYakushev 10:53, November 6, 2011 (UTC)