Farther and further both mean at a greater distance, and they are used interchangeably in this sense. In the United States, though, farther is more often used to refer to physical distances, and further more often refers to figurative and nonphysical distances. For example, we might say that one mountain is farther away than another, while we might say the price of a stock (a nonphysical thing) fell further today than yesterday. This is not a rule, however, and further is often used for physical distances. The distinction does not exist in the U.K. and elsewhere in the (British) Commonwealth of Nations, where further is preferred for all senses of the word and farther is rare.
Further has senses it does not share with farther. It works as an adjective meaning additional—e.g., “I have no further questions.” It works as an adverb meaning additionally—e.g., “He said he did not spend the money, and stated further that he had never even received it.” And it works as a verb meaning to advance (something)—e.g., “This website is meant to further understanding of 21st-century English.” Farther is not commonly used these ways.
The physical/nonphysical distinction in the U.S. extends to the superlatives farthest and furthest. Furthermore is an adverbial extension of further and often bears replacement with the shorter word. The rare furthermost is sometimes used to mean farthest or furthest, and it likewise bears replacement with the shorter words.
In these sentences, farther refers to physical distances:
Farther south, in Central Otago, there are some even harsher mountains. [Telegraph]
Making people park a little farther away will actually increase their exposure to danger, he added. [Red Wing Republican Eagle]
Pyzik said the testing cells would be located farther from the school than initially proposed. [Chicago Tribune]
And in these examples, further refers to figurative and non-physical distances—for example:
The Dollar is extending its gains against the Euro. EUR/USD fell further to 1.3430, hitting a fresh daily low. [NASDAQ]
So the mechanism that was initially meant to be protective can become the source of further damage. [Indy Posted]
Reliable measurements of the Sun’s magnetic field are only available from 1900 onwards, so researchers used computer simulations for further back in time. [Daily Mail]
Many counterexamples could be found, however, and using further in place of farther is never an error.