Well-nigh means almost, or very nearly. It goes in front of an adjective or a word like impossible, and it says that something is so close to being true that the difference hardly matters: the job is well-nigh impossible, the old house was well-nigh empty, it was well-nigh dark by the time we got home.

The thing to know is that this word is old. Nobody in Taiwan needs to say it, and if you say it to a friend you will sound like a character in a novel. But you will meet it in reading — in older books, in newspaper writing that wants to sound a bit grand, and sometimes in a speech. Writers still reach for it because it sounds heavier and more final than plain almost.

It likes to sit next to strong, absolute words. Well-nigh impossible, well-nigh perfect, well-nigh unbearable are the ones you see again and again. Nobody says well-nigh warm or well-nigh happy — it wants a word that already means the very end of the scale, and then it pushes you right up to that end without quite touching it. So well-nigh impossible does not mean impossible. It means: not quite impossible, but do not expect it to work.

A note on writing it: you will see it with a hyphen (well-nigh) and sometimes as two words (well nigh). Both turn up in print. The hyphen is the safer choice.