Because the puzzles are auto-generated and essentially random, it’s really difficult to generate them with a particular difficulty. Instead, after generating the board, Dominate tries to solve the puzzle itself using a human-like heuristic approach, and gives you a target time to solve the puzzle in. Harder puzzles will take longer than easier ones.
The time that it generates is then adjusted up or down according to the level you are playing at. Essentially, the level you gain is there to give you an idea as to how quick you are at solving the puzzles. You can choose the level you are playing at from the Settings screen, where you can also see your performance relative to the target times you have been set.
Dominate will also automatically progress you to the next level when you are consistently beating your target times. This happens when you have played at least ten games at a your current level, and the times for your most recent five games are on average 75% of the target time.