This is one of the few usage rules where American and British English genuinely diverge, rather than one having drifted from the other.
Used for physical, measurable distance.
Used for figurative distance, degree, or the sense of "additional." Also a verb meaning to advance something.
Traditionally, farther is for physical distance you could measure, and further is for everything figurative: extent, degree or something additional. You walk farther down the road, but you look into a matter further, and you ask further questions.
In practice, further has been taking over. Many writers now use it for distance too, and that is widely accepted, especially in British English. Farther has not spread the other way: it still means only distance. So further is never wrong for distance in casual writing, but farther is wrong for anything that is not distance.
For careful American writing, the simplest rule is this: if you could replace the word with a number of miles or feet, farther is the tidy choice. If it means more or additional, it must be further.
Do you have any farther questions?
✓Do you have any further questions?
This means additional questions, not distance, so only further works.
We need to discuss the budget farther before we sign.
✓We need to discuss the budget further before we sign.
Discussing something more is a matter of degree, not distance: further.
Nothing could be farther from the truth.
✓Nothing could be further from the truth.
The distance here is figurative, so the traditional choice is further.
Quick way to remember
If you could measure it in miles or metres, use "farther" — both contain "far." For anything abstract, additional or advancing, use "further." When you genuinely cannot tell which, "further" is the safer choice; it covers both in British usage and reads as natural in American.
Can further be a verb?
Yes, and farther cannot: "she took the course to further her career." It means to advance or promote.
Is it "further to my email"?
Yes. It is a formal way of saying "following on from my email", and it always uses further.