Use either farther or further as an adjective for literal distance (The airport is farther [further] than I had figured), but also use further to mean additional, as in She gave a further opinion that evening. (The adjective further used to be limited to such figurative uses, just as farther continues to be applied only to literal distance.) As adverbs, farther and further are nearly interchangeable today, although further is much more frequently used: We walked further [occasionally farther] today than ever before. He expanded further [rarely farther] on his original proposal. But use only further as a sentence adverb: Further, he insisted that we stay overnight. (Some commentators predict that further will one day overwhelm farther in all uses except the adjective meaning literal physical distance.) See also ALL THE.
Farther and further are comparatives, both based ultimately on the adjective far. Farthest and furthest are the superlatives: This is the farthest [furthest] point reached by the paved road. Quitting was the thought furthest [farthest] from my mind. These uses are Standard; the forms fartherest and furtherest are Vulgar English and Substandard.