Jacob Taylor, Thane Krios, and Garrus Vakarian are possible male characters to "Romance" with.
Miranda Lawson, Subject Zero; Jack, and Tali'Zorah nar Rayya are possible female characters to "Romance" with.
Talk to the person you want to romance with often and/or after every mission you do. Just try not to be an ass to them, play nice. Make sure you do their loyalty mission. When you go through the Omega 4 Relay a cut-scene will start with that person if done right.
And remember, Safety First!
Note: For Jack, Romancing with her before the Omega 4 Relay using your Renegade will not get this achievement.
EDIT: BDSLiVe said, That you cannot do same sex romance.
EDIT: I personally have never tried but I have seen videos on Youtube of Mass Effect 2 same sex romances. But either way, you can be the test subject and let me know.