I understand that the hammer applies a force on the nail (of magnitude say, 25 N). The force is then transmitted through the nail onto the wood and thus, the nail exerts a force of 25 N on the wood.

But then according to Newton's third law of motion, the wood would apply a force on the nail (of mag. 25 N).

In theory, this force on the nail by hammer (25 N) and the force on the nail by wood (-25 N) must cancel each other and the nail shouldn't move (supposing it was at rest when it was hammered).

However, in reality, the nail does move and gets embedded. How is this possible?