Image text: The top computer champion at Seven Minutes in Heaven is a Honda-built Realdoll, but to date it has been unable to outperform the human Seven Minutes in Heaven champion, Ken Jennings.

This explanation got really long, so I'm placing it after the jump.

Ok, let's get right into it.  To understand the comic, you have to understand what the games are, so let's go (but first, the years in parenthesis in the comic are the year that the game was mastered by a computer):

Tic-Tac-Toe - (via wikipedia) Tic-tac-toe, also called noughts and crosses (in the UK, Australia, New Zealand, South Africa, India and the rest of the British Commonwealth countries), is a pencil-and-paper game for two players, X and O, who take turns marking the spaces in a 3×3 grid.

Nim - (via wikipedia) a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap.
Ghosts - I'm going to guess this board game (via Board Game Geek), but the commenters have told me that it could also be this spoken word game.

Connect Four - (via wikipedia) (also known as Captain's Mistress, Four Up, Plot Four, Find Four, Fourplay, Four in a Row and Four in a Line) is a two-player game in which the players first choose a color and then take turns dropping their colored discs from the top into a seven-column, six-row vertically-suspended grid.

Gomoku - (via wikipedia) an abstract strategy board game. Also called Gobang or Five in a Row, it is traditionally played with go pieces (black and white stones) on a go board (19x19 intersections); however, because once placed, pieces are not moved or removed from the board, gomoku may also be played as a paper and pencil game. This game is known in several countries under different names.

Black plays first, and players alternate in placing a stone of their color on an empty intersection. The winner is the first player to get an unbroken row of five stones horizontally, vertically, or diagonally.

Scrabble - (wiki) a word game in which two to four players score points by forming words from individual lettered tiles on a gameboard marked with a 15-by-15 grid.

Counterstrike - is an first person shooter video game. That doesn't make much sense does it?  The commenters suggest that it could be this backgammon variant. (via Board Game Geek)

Beer pong - (via wiki) also known as Beirut, is a drinking game in which players throw a ping pong ball across a table with the intent of landing the ball in a cup of beer on the other end.
Here's the video of the University of Illinois robot mentioned in the comic.

Reversi - (via wiki) (also marketed by Pressman under the trade name Othello) is a board game involving abstract strategy and played by two players on a board with 8 rows and 8 columns and a set of distinct pieces for each side. Pieces typically are disks with a light and a dark face, each face belonging to one player. The player's goal is to have a majority of their colored pieces showing at the end of the game, turning over as many of their opponent's pieces as possible.

Chess - (wiki) Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. Each player begins the game with sixteen pieces: one king, one queen, two rooks, two knights, two bishops, and eight pawns, each of these types of pieces moving differently.

Jeopardy - (wiki) an American quiz show featuring trivia in history, literature, the arts, pop culture, science, sports, geography, wordplay, and more. The show has a unique answer-and-question format in which contestants are presented with clues in the form of answers, and must phrase their responses in question form.
Ken Jennings, mentioned in the image text is a famous Jeopardy champion.

Starcraft - (wiki'ed) a military science fiction real-time strategy video game... the game revolves around three species fighting for dominance in a distant part of the Milky Way galaxy known as the Koprulu Sector: the Terrans, humans exiled from Earth skilled at adapting to any situation; the Zerg, a race of insectoid aliens in pursuit of genetic perfection, obsessed with assimilating other races; and the Protoss, a humanoid species with advanced technology and psionic abilities, attempting to preserve their civilization and strict philosophical way of living from the Zerg.

Poker - (wiki) a family of card games involving betting and individualistic play whereby the winner is determined by the ranks and combinations of their cards, some of which remain hidden until the end of the game.

Arimaa - (wiki) a two-player abstract strategy board game that can be played using the same equipment as chess. Arimaa was designed to be more difficult for artificial intelligences to play than chess. Arimaa was invented by Omar Syed, an Indian American computer engineer trained in artificial intelligence. Syed was inspired by Garry Kasparov's defeat at the hands of the chess computer Deep Blue to design a new game which could be played with a standard chess set, would be difficult for computers to play well, but would have rules simple enough for his then four-year-old son Aamir to understand.

Go - (wiki) an ancient board game for two players that originated in China more than 2,000 years ago. The game is noted for being rich in strategy despite its relatively simple rules. The game is played by two players who alternately place black and white stones on the vacant intersections (called "points") of a grid of 19×19 lines (beginners often play on smaller 9×9 and 13×13 boards). The object of the game is to use one's stones to surround a larger portion of the board than the opponent.

Snakes and Ladders - (via wikipedia) (or Chutes and Ladders) an ancient Indian board game regarded today as a worldwide classic. It is played between two or more players on a gameboard having numbered, gridded squares. A number of "ladders" and "snakes" (or "chutes") are pictured on the board, each connecting two specific board squares. The object of the game is to navigate one's game piece from the start (bottom square) to the finish (top square), helped or hindered by ladders and snakes, respectively.

Mao - (via wiki) (or Mau) a card game of the Shedding family, in which the aim is to get rid of all of the cards in hand without breaking certain unspoken rules. The game is from a subset of the Stops family, and is similar in structure to the card game Uno or Crazy Eights.
The game forbids its players from explaining the rules, and new players are often told only "the only rule you may be told is this one." The ultimate goal of the game is to be the first player to get rid of all the cards in their hand.

Seven Minutes in Heaven - (yes, seriously I went to wikipedia!) a teenagers' party game first recorded as being played in Cincinnati in the early 1950s. Two people are selected to go into a closet or other dark enclosed space and do whatever they like for seven minutes. Sexual activities are allowed; however kissing and making out are more common.

And finally Calvinball is a reference to the comic strip Calvin and Hobbes by Bill Watterson - (via wikipedia)

Calvinball is a game played by Calvin and Hobbes as a rebellion against organized team sports; according to Hobbes, "No sport is less organized than Calvinball!" Calvinball was first introduced to the readers at the end of a 1990 storyline involving Calvin reluctantly joining recess baseball. It quickly became a staple of the comic afterwards.

The only hint at the true creation of the game ironically comes from the last Calvinball strip, in which a game of football quickly devolves into a game of Calvinball. Calvin remarks that "sooner or later, all our games turn into Calvinball," suggesting a similar scenario that directly led to the creation of the sport. Calvin and Hobbes usually play by themselves, although in one storyline Rosalyn (Calvin's baby-sitter) plays in return for Calvin doing his homework, and plays very well once she realizes that the rules are made up on the spot.

The only consistent rule states that Calvinball may never be played with the same rules twice. Scoring is also arbitrary, with Hobbes at times reporting scores of "Q to 12" and "oogy to boogy." The only recognizable sports Calvinball resembles are the ones it emulates (i.e., a cross between croquet, polo, badminton, capture the flag, and volleyball.)

Image text: We're going to have to work together to get over our hangups if we're going to learn to move on Catan's hexagonal grid. It's bad enough that we lost our crew of pawns when we passed within firing range of Battleship.

In this comic appears to be two chess pieces, a knight and a bishop.  The knight can only travel two spaces up or over and then one space over and the bishop can only travel diagonally.  The chess pieces appear to be on a 3x3 chess board on wheels that has broken off from a "home board".

The chess pieces are playing Settlers of Catan, which is the reference to the hexagonal squares because that's the shape of the spots on the board.

Ba3, Nc3 and Ke5 are all moves on a chess game.  The black squares joke is because the bishop can only go back and forth diagonally on the white squares.

There is also a reference to another board game, Battleship in the image text.

Image text: The only winning move is to play, perfectly, waiting for your opponent to make a mistake.

Ok, wow.  I don't know if I ever though I'd see a tic-tac-toe decision tree laid out like this.  The explanation at the top sums it up for how to find your next move.  Click on the image to see it large so you can find your next move.  And as usual, the image text has it perfectly because the only way to win tic-tac-toe is to have the other person screw up somehow.

Image text: 'How about a little ... *family growth*?'  'Dude, that's not until round two.'

This comic starts out with a cliched pornographic movie set up.  A strong pizza delivery guy mistakenly enters a house where a French maid is dressed in what appears in the comic to be lace.  The pizza delivery got has a "hot sausage" pizza, in which the hot sausage is a double entendre for his penis.

Then in the second frame, the pizza delivery guy turns the pizza sideways which ruins all chances he had of ever delivering that pizza.  Literally, I'm not making another sexual innuendo there.

Then the second cliche porn set up enters the picture as a plumber enters then delivers his sexual inneundo that Mrs. Jones "needs her plumbing fixed".  But, then because this is xkcd instead of turning into some wild Pay-Per-View porn movie, they discover the game Agricola and decide to play that instead.

Agricola is a turn based game in which each player is a farmer and spouse.  Each player is dedicated to expanding their farm and accomplishing all the tasks on the farm.  This is where the sexual innuendo comes in from the image text.  "Family Growth" is obviously a sexual innuendo, but in Agricola you cannot complete actual family growth until after you expand your farm and farm house to accommodate the kiddies.

In the last frame we see Mr. and Mrs. Jones come home.  You can tell Mr. Jones because he has the briefcase.  When they come home to interrupt the game, the French Maid, the Pizza Delivery Guy and the Plumber are all playing Agricola on the floor.

Image text: At one point, by force of childhood habit, the doctor accidentally removed three or four organs.

Operation is a "board" game in which the player attempts to remove body parts from holes in the board without touching the metal sides of each hole.  Operation is hooked up to a buzzer which emits the familiar "Bzzzzt!" sound whenever the removal instrument touches the sides.

Image text: You roll for initiative, and ... [roll] ... wow, do you ever take it.

In this comic, there are two different groups playing two different dice games.  However, the dice got mixed up.  Instead of numbers, the first two are supposed to have another die with actions on them, as seen in the last panel.  The "sex" die got mixed in with the kids Dungeons and Dragons set because they did not organize the game cupboard.  The four people in the last panel are completely different than the two in the first 3.

The joke in the image text is a play on the word "initiative".  In Dungeons and Dragons you roll for initiative for an extra modifier to your roll of the dice.  In the image text, they are suggesting that someone accidentally rolled a sex die for initiative.  And "take initiative" means to take charge to take command outside of Dungeons and Dragons.

Image text: HOW ABOUT A NICE GAME OF STRIP GLOBAL THERMONUCLEAR WAR?

Agricola is a board game in which you start out as a farmer with a spouse.  It is a turn-based game in which you have two possible movements for each character you possess.  Sounds enthralling...

Jumanji is the game played in the movie by the same name.

Poohsticks is a game played in the Winnie the Pooh books in which two "players" each drop sticks from a bridge and the first stick to make it to the end wins.  Sounds enthralling as a strip competition...

Podracing is the type of racing featured in Star Wars Episode I.

Iterated Prisoner's Dilemma is a problem in game theory.  It is the case that if two prisoners are taken into jail, but kept separate.  If both choose to remain silent, they are given 6 months jail time.  If they both accuses the other, they both will do 5 years in prison.  If one accuses the other while the other stays silent, one goes to jail for 10 years and the other gets away scot free.  When it is iterated, it is played over and over again.

Chess by Mail is just what it sounds like... and very slow.

Conway's Game of Life is a zero-player game and played by cells and here are the rules from Wikipedia:

1. Any live cell with fewer than two live neighbours dies, as if caused by underpopulation.
2. Any live cell with more than three live neighbours dies, as if by overcrowding.
3. Any live cell with two or three live neighbours lives on to the next generation.
4. Any dead cell with exactly three live neighbours becomes a live cell.