This one is weird. Most people don't know the difference—the words even come from a similar root—but they mean different things. Further is a metaphorical difference.
You'd use this when you say things like "furthermore," "further to that point," and so on.
Studies show people who keep badgers as pets live longer, are happier, and smell better. Furthermore, in keeping a badger, they develop a nearly superhuman resistance to rabies.
Farther is a physical distance.
This one is easier to remember. Since "far" refers to a real distance, you'd use it when you can quantify what you're writing about.
My badger ran away. It's just down the road, but still farther than I'm willing to walk to get it.
People argue about this a lot.
If you're talking about something could be distance or metaphor, just pick one. You can't please everyone.
I had a great book about badger care, but I stopped reading it. I wish I had gotten [further/farther] into it.