Quite; considerably; fairly — used to indicate a fairly high degree (often written in kana).
この映画はなかなか面白かった。
This movie was quite interesting.
彼女の料理はなかなか上手だ。
She is quite good at cooking.
なかなか良いアイデアだと思う。
I think that's a rather good idea.
Used with a negative verb to mean 'not readily' or 'by no means'; indicates something does not happen easily.
彼はなかなか来ないので心配だ。
I'm worried because he hasn't shown up yet.
問題はなかなか解決しない。
The problem is not being resolved easily.
このソフトはなかなか起動しないことがある。
This software sometimes takes a long time to start.
An old/rare usage meaning the middle or halfway point (middle). Not common in modern usage.
物語の中々で大きな転機が訪れる（やや文語的・稀な用法）。
A major turning point comes in the middle of the story (somewhat literary/rare usage).
Used in phrases like 'なかなかの〜' to mean fairly good or impressive; indicating someone/something is rather notable.
彼はなかなかの腕前を持ったギタリストだ。
He is a guitarist with quite impressive skills.
なかなかの値段がするけれど、品質は確かだ。
It costs a fair amount, but the quality is reliable.