Will all 21 million units be completely mined one day?
Theoretically yes, but that moment is so far in the future it is likely that some other events such as new currency technology will surpass bitcoins.

However, mining will not stop.  Mining is the security backbone of bitcoin, and it will continue because it will still be profitable to mine.  Miners will be paid fees from existing bitcoins with every transaction instead of a reward of newly created bitcoins.