Here is a small recap from the website.
http://www.cwtv.com/shows/gilmore-girls/episodes/770
With graduation looming, Rory (Alexis Bledel) is anxiety-ridden over her future, so Lorelai (Lauren Graham) tries to cheer her up with a day of eating and shopping. Lorelai notices happily that Luke (Scott Patterson) is once again wearing the baseball cap she gave him, and invites him to join her and Rory for karaoke night at a local bar. After a few drinks, Lorelai takes the stage to sing a heartfelt number which is obviously directed at Luke.
Logan returns from a business trip to California and comes to see Lorelai to discuss his future with Rory.
The song was..I will Always Love you and Logan wants permission to marry Rory.