From the Mass Effect Wiki (which is derived from the codex of the three games):
Although asari have one gender, they are not asexual. An asari provides two copies of her own genes to her offspring, which—regardless of the species or sex of the 'father'—is always an asari. The second set is altered in a unique process called melding, also known as the joining.
During melding, an asari consciously attunes her nervous system to her partner's, sending and receiving electrical impulses directly through the skin. A common phrase used before melding is "embrace eternity," presumably to help focus the partner's mind. Effectively, the asari and her partner briefly become one unified nervous system.
The partner can be another asari, or an alien of any gender. However, since the asari began encountering other sentient species, non-asari mates have become preferred for the diversity they provide. This reproductive process can lead to some confusion among non-asari, who might expect offspring with "mixed" genes. However, the offspring is always 100% asari as no DNA is taken from the partner, but is rather used as a "map" to randomize the genes of the offspring.