Somebody's already described rugby, so I'll tackle AFL.
How long do the games last? Are they divided into quarters, halves, etc?
A game consists of four 20 minute quarters; the clock is stopped when someone scores or the ball goes out of bounds. Note that the official game time is known only to the umpires; the timer you see on TV or in the stadium is only an estimate.
How big is the field / how many people are in play?
Believe it or not, required dimensions for an AFL field are not codified. As a result, different grounds have different sizes. For grounds used by teams in the AFL competition, lengths vary from 151m - 179m, and widths from 115m - 150m. Each team fields 18 people.
What basically goes on during the game?
You score by kicking the ball through the goalposts at the ends of the ground. Kicking the ball through the two tall posts yields six points (a goal); kicking it between a tall and short post yields one point (a behind). If the ball is touched by someone after being kicked, but prior to passing through the goalposts, one point is awarded, regardless of which posts the ball is kicked through. Hitting the tall posts with the ball also yields one point. After a goal is scored, the game is restarted by the umpire in the middle of the ground; after a behind, the ball is kicked into play from the goals.
The ball is passed by either kicking or handballing to another team member. A handball is basically where you hold the ball in the palm of one hand, and punch it with the other; throwing the ball is against the rules. Once a player obtains possession of the ball, they can run with it, but must bounce or touch it on the ground every 15 metres. If a player catches a ball that has been kicked over 15 metres, they are awarded a mark and may take a set shot.
A player in possession of the ball can be tackled. This is done by grabbing hold of them between the knees and shoulders. They must then get rid of the ball, else be penalised. Tackling below the knees or above the shoulders is against the rules, as is pushing someone in the back.
When the ball goes out of play, it is thrown in by an umpire. However, if a player kicks the ball and it goes out of play without bouncing or being touched, it is 'out on the full' and a free kick is awarded to the other team.
What are the big competitions? Who are the "best" teams?
The AFL is the national league, and the biggest by a huge margin. Each state also has their own league, such as the WAFL in Western Australia. The AFL actually evolved out of the VFL (the Victorian state league, which still exists separately), as a result, ten of the eighteen teams in the AFL are from Victoria, with Western Australia, South Australia, Queensland, and New South Wales providing two teams apiece.
The current premier (grand final winner) are Hawthorn, who won their third-consecutive final in 2015. Geelong and Sydney have also recently been well performing.
Who are the most famous players in history?
This really depends on who you're asking and where they live. Naturally, players who are household names here in Western Australia will not be as famous in Victoria, and vice-versa. Nationally, some of the better known ones are Tony Lockett (retired, leading goalkicker in the competition) and Gary Ablett, Sr. (and his son). Some very famous players were around before the national competition existed, and played for state teams (for example Barry Cable in the WAFL).