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.
Luigi had been practing his French skills at Mario Brothers Plumbing in order to attract women, to the point where Mario had gotten sick of it. Unable to stand Mario's griping, Luigi left to practice at the park. At the park, he met up with a woman who fell in love with him due to the fact that he could speak French.
Later, Luigi brought his girlfriend to Mario Brothers Plumbing, where he praised her for her beauty, while she praised him for his French skills. Meeting up with Mario, Luigi's girlfriend asked him if he spoke French, to which he answered no. Uninterested in Mario's lack of French, she turned back to Luigi, asked him to show her his French etching, and the two of them walked off.
Luigi's girlfriend never appeared again, implying that their relationship probably only lasted for a very short length of time.