"Farther" and "further" are comparative adverbs or adjectives. They are the irregular comparative forms of far. We use them to talk about distance. There is no difference in meaning between them. "Further" is more common:
We can’t go any further; the road’s blocked.
"Farther", and, much less commonly, "further" can be used as adjectives to refer to distance away from the speaker:
He could see a small boat on the farther shore.
At the further end of the village stood an old ruined house.
We often repeat "farther" or "further" to emphasise the distance:
‘I am just a little ship,’ Aunt Emily said, ‘drifting farther and farther out to sea.’
We often use along with farther and further:
Ben Gunn had told me his boat was hidden near the white rock, and I found that rock farther along the beach.
We often use a little, a bit or a lot before further and farther:
[in an aerobics exercise class]
Now push and stretch that arm just a little further and count to ten.
There are some occasions when we can use further but not farther.
We use further before a noun to mean ‘extra’, ‘additional’ or ‘a higher level’:
For further information, please ring 095-6710090.
She’s gone to a college of further education. (a place to study practical subjects from age 17)
We also use further to mean ‘more’:
I do not propose to discuss it any further.
The expression "further to" is often used in formal letters and emails when someone writes as a follow-up to a previous letter or email:
Further to my email of 22nd January, I’m now writing to ask if you have considered our offer and whether you wish to proceed with the contract.