In Influence, you find yourself placed on a three-by-three set of discs. These discs are known as cells. Cubes have fallen onto the cells. Your goal is to clear the cells of these cubes.
This is where your special power known as "Influence" comes into play. Influence is used to clear cubes. You have access to multiple levels of Influence, and you can freely choose which level you want to cast and the cell from which you want to cast it.
The game also has a somewhat unique -- unique for a puzzle game, at least -- way of adding extra difficulty. You have access to three different characters. Depending on the character, the difficulty changes.
These characters have stories. Fay is a medical researcher who began having strange dreams after she started dating a guy a few years back. Tanaka is a diplomat with language abilities in Russian, English and Japanese. Influenced by his grandfather, he has an ambition of some form. The third character has to be unlocked by clearing the game.