What episode is it that Rory and Lorelai get into a fight and Rory ends up running away to her grandparents' house?

There were actually two different times. I'm not really sure which time you are refering to though. The first time was in the Season 1 episode "P.S. I Lo..." where Rory runs away to Emily and Richard's house for only one night. That was because Lorelai hadn't told Rory that she was back together with Max. The other time was in Season 5 "A House is Not a Home". This was the infamous falling out that was caused when Rory dropped out of Yale. Thus resulted in Rory and Lorelai being seperated for "5 months, 3 weeks, and 16 days"! Hope that helped:)

The last episode of season 5.

It's that episode where Rory first contracts a viral disease in her intestines. Her granparents die in the newest episode