MAME: Pac-Man

Created by Namco in 1979, Pac-Man is one of the enduring classic arcade games - it is regularly in the top 10 games downloaded for mobile phones, and has spawned many spin-off sequels -- the most original being, in my opinion, Pac-land (1984, arcade), Pac-Man Vs. (2003, GameCube - in which the ghosts are controlled by other players) & Pac-Pix (2005, DS - where Pac-Man is literally drawn on the screen by the player).

Gameplay
Pac-Man takes place in a maze-like screen. The passages are filled with small, glowing yellow dots which must all be collected to advance to the next level. There are also 4 larger "power pellets" in each corner, which I'll get to later.
Pac-Man's player avatar is a very simple shape -- a day-glo yellow character which alternates between a circular, to circular with a wedge-shaped hole to represent Pac-Man's ever-opening mouth. He is controlled with a four-direction joystick (or in MAME, 4 keyboard keys).

To create challenge, 4 coloured ghosts roam the mazes & attempt to tag the player's Pac-Man character. They each have their own chase-style, which experienced players are known to exploit to never get caught.
If the player collects a "power pellet", all ghosts will turn dark blue, and their expression will change from their normal searching eyes, to expressions of fear -- if they are eaten by the player in this state, their eyes will humourously to the middle of the maze & the ghost will return, back in its regular colour. (After c. 20levels, ghosts cannot be eaten).
Also, occasionally fruit will appear in the middle of the maze, which can be collected for more points.
When all the dots in a level are collected, the maze refreshes, and the player plays the same level again, only with faster ghosts.

Final view
Pac-Man's cartoony presentation helps it gain a lot of character, and also makes it seem very friendly to non-regular games players. It's very simple gameplay (control is purely by joystick) makes it very easy to get into, while mastering it requires hours of dedication to realise the patterns necessary for survival at the later levels' faster speeds.
Classic.

MAME: Asteroids

Asteroids
Asteroids was one of the founding fathers of the shoot-em-up genre.
Elegantly presented, Asteroids is drawn using Vector graphics instead of sprites. It can be imagined that its classy, white-on-black graphics would have glowed through any darkly-lit arcade as was common at the game's release (1979).

Gameplay
The player's control consist of clock- and counter-clockwise turning, a fire button, a thrust button & a hyperspace button. To slow down, it is necessary to turn and fire thrust in the opposite direction to one's momentum. As the gameplay takes place on a closed-in space, leaving the area's borders will cause the ship (Spacewar!'s classic wedge) to reappear at the opposite side of the screen. The hyperspace button will teleport the player's ship to a random location on the screen, with the danger of either random destruction, or being in the path of an incoming asteroid.
These asteroids are created at the beginning of each level (starting with 4, increasing by 2 each level to a max of 12). Each asteroid, when shot, will break up into 2 medium-sized rocks, with each of these breaking up into 2 smaller ones before being finally vanquished. Additionally, the smaller sizes move at faster speed than the previous ones. This is dangerous for the player, as one hit will destroy his ship & lose him a life.
Finally, UFO's (shaped like flying saucers) will periodically appear. They come in 2 sizes: larger ones move slowly & fire in a random pattern, whereas smaller UFOs will attempt to actively hunt the player's ship. They will disappear if the player doesn't destroy them within c. 10 seconds.
Like the player, asteroids & UFO's will wrap around the screen.

Final view
Asteroids is a great game. It makes the player have to continually assess where his momentum will take him, what angle his shots must be at to hit that asteroid in relation to where it will be in 2 seconds & whether he will be able to thrust out of the way of the incoming shrapnel, or have to use hyperspace, or whether he can win a firefight with that damn small UFO -- in other words, the game is challenging without being unfair, and allows for continuous improvement.

Chess

Chess is a very deep strategy game -- in fact, it's been mathematically calculated that there are more possible chess games than there are atoms in the universe!
Now that the drama's outta the way...

Chess, in its current form, is played on an 8x8 grid, with squares alternating between black & white. Both players set up their pieces at the beginning of the game -- one row of pawns, the other with king, queen and 2 each of rook, knight & bishop.

Starting with white, players take turns moving one piece at a time in an attempt to capture their opponent's king, by means of positioning their own pieces into a position from which the king cannot escape. This requires deep strategy, involving sacrifice plays (allowing the opponent to capture a piece to gain an advantage for oneself), stalemate plays (protecting one piece with another to discourage capture) & forced-choice plays (positioning pieces so that the opponent must choose which piece he'd rather sacrifice). Due to white's first turn, he is usually in advantage, usually forcing the black player to play to a tie instead of a win.
Of course, new players (and those playing without good strategy) will just try to capture as many pieces as possible in the hopes of forcing the opponent to concede due to lack of force.