In terms of the damage of a single attack the highest are: Solarbeam and Hyperbeam both at 120. However remember that this damage does not mean it is the best move because both are special attacks that require 100 energy and they have very high delay on the attacks.
If you want to see the moves with the highest DPS (damage per second) checkout this Google Spreadsheet. DPS is a better measure on which attack is the best. Note that after Niantic's huge damage update, most of the moves are relatively close in terms of DPS, so types matter more than just pure DPS. Also, after this update the special attacks were buffed a lot so they are definitely worth using instead of spamming the standard attack.
The best moves for attacking are:
- Pound for Standard Attack at 13 DPS with attack speed of 540 ms
- Stone Edge for Special Attack at 32 DPS with attack speed of 3100 ms
The best moves for defending are:
- Bubble for Standard Attack at 6 DPS
- Stone Edge for Special Attack at 20 DPS
Credits for the spreadsheet goes to this Reddit Post.
UPDATE: This post has been updated after Niantic's huge update to the DPS of moves (July 29, 2016).