To earn this achievement you’ll need to strike up a romantic relationship with one of your squadmates. Your options in Mass Effect 1 are fairly limited; Liara is interested in Shepard regardless of their gender, while Kaiden and Ashley are only interested in heterosexual relationships.
Regardless of whom you choose to romance, establishing a romantic relationship is remarkably similar. Talk to the character you’re interested in after each core mission (Therum, Feros, Noveria and Virmire) and generally stick to the upper right dialog options. If you flirt with more than one character, they’ll eventually start questioning you about their rival - if you express interest in their rival or disinterest in the character you’re talking to, you may end up sabotaging the romance. If you continue to flirt with two characters, at some point you’ll be confronted by them and have to pick one.
You’ll know you’ve successfully set up a romance after you complete all four of the aforementioned core missions, as your romantic interest will try to console you when the Normandy is grounded at the Citadel. The achievement won’t unlock there, but it should unlock shortly thereafter when you travel to Ilos.