These two English words are very similar; keep reading for further information.
Farther means "more far/distant" in physical space.
Tom ran farther than Bill.
Which is farther, New York or Los Angeles?
Farthest means "the most far/distant."
I prefer the seat farthest from the window.
Pluto was once considered the farthest planet from the sun.
Further means "more far/distant" figuratively or non-physically, and can also mean "more/additional":
Nothing could be further from the truth.
Who is further along in her research?
Do you have any further ideas?
Further is also a verb: "to advance":
This class will further your understanding of ethics.
Furthest means "the most far/distant" figuratively.
That is the furthest thing from my mind.
The Bottom Line
Many English speakers use both farther and further to mean "more far" in space, but only further can be used for non-physical distance and to mean simply "more." If you are talking about actual distance, you can usually use either word without raising any eyebrows, but keep further in mind for all of your English furthering efforts.