Throughout the game, you can talk to all of your squad members and learn about them. Certain squad members, if you talk with them enough, you can eventually romance. Who you can romance depends on the gender you choose to be, but the romance options are as follows.
Male Shepard: Ashley Williams, Liara T'Soni
Female Shepard: Kaidan Alenko, Liara T'Soni
Whoever you decide to romance, make sure you talk with them after every story mission, and always choose paragon or neutral choices so as to keep the romance going. Even after they express interest, nothing will happen until the end of the game. Once you finally are able to go to Illos, your romantic interest will approach you. Continue to be supportive and eventually you will be given the option to invite them in for the night. Once the scene plays out, you will unlock the achievement.