From the Super Mario Wiki
Luigi's girlfriend is an unnamed woman whom Luigi befriended for a brief period of time. Little is known about her, except that she adores men who can speak French. She appears in the live-action The Super Mario Bros. Super Show! segment "French".
Luigi has been practicing his French skills at Mario Brothers Plumbing in order to attract women, to the point where Mario gets sick of it. Unable to stand Mario's griping, Luigi leaves to practice at the park. At the park, he meets up with a woman who falls in love with him due to the fact that he can speak French.
Later, Luigi brings his girlfriend to Mario Brothers Plumbing, where he praises her for her beauty, while she praises him for his French skills. Meeting up with Mario, Luigi's girlfriend asks him if he spoke French, to which he answers no. Uninterested in Mario's lack of French, she turns back to Luigi, asks him to show her his French etching, and the two of them walk off.
Luigi's girlfriend never appears again, implying that their relationship probably only lasted for a very short length of time.