Player is a term used for a person playing a video game. The first Mario game, Donkey Kong, was originally first played and tested by Shigeru Miyamoto and a few of his employees, trying out the game and at the same time seeing who could beat it first. Although most players are human, Mario media has also shown the series' specific species to enjoy certain video games, which are usually fictional.
In some games, mostly the RPG games, many characters break the fourth wall and talk to the player ("you in front of the TV"); these characters include Goombella, Lord Crump, and Professor Frankly from Paper Mario: The Thousand-Year Door. It is also worth noting that several characters use Mario (or whatever playable character) as a device to talk to the player; for instance, Mario's RPG adventures often show supporting characters giving Mario commands like "press the A button". These instructions are actually directed to the player. Super Paper Mario makes a point out of this by having Mario ask what the person exactly means by "press the A button".
In Super Paper Mario, characters such as Bestovius refer to the player as a deity-like being existing within another dimension. Thoreau describes the player as the great being who watches them when Mario asks what the " Button" is. It may be worth noting that the manual of Super Mario Bros. 2 explains the dream-based origin of certain enemy characters (for instance, it establishes that Cobrats come from Toad's dreams), and it happens to state that Ninjis "appear in the dreams of NES players". Another reference to the player is in Hotel Mario, in which there are many occurrences of Mario, Luigi, and Peach breaking the fourth wall. Some examples include Mario telling the player to check out the enclosed instruction book for help, Luigi saying "And you gotta help us", and Peach saying "And I thank you too". At the end of Hotel Mario, Mario, Peach, and Luigi all exclaim "You're the best player ever!".