Australian versions of "football" include:
Football (what we call soccer): This isn't football...it's soccer. Kick the ball down the field to score goals; don't touch it with your hands; no tackling; padding unnecessary.
Australian Rules Football (or "Footy"): Soccer + Basketball-like game played with a ball that looks like a Football on a modified Cricket field. Kick the ball between the goal posts to score; bounce the ball or touch it to the ground intermittently so you don't get called for holding; no throwing the ball; same contact as soccer or basketball - no tackling; no padding required.
Rugby: Football + Wrestling + (in the case of Rugby Union) Cheerleading. Run the ball down the field to score a "try" (like a touchdown) and pick up the extra point by kicking the ball through the posts (like a field goal - not sure what this is called). (I have to say, I love the term "try" - they passed out signs to everyone at the first rugby match I went to in Queensland that said "TRY!" on them and I felt like I was cheering at a little kid's game where parents were promoting good effort over winning.) No pads - just a mouthguard and an optional piece of fabric over your head to keep your ears from getting ripped off during the wrestling portion of the game. (A few guys had on what looked like a kneepad or brace, but I assume this was just holding their leg together.) Players beat down the possessor of the ball, then, without a stop in gameplay, they scramble up, pass it off to a teammate, and keep pushing down the field. After they wrestle the ball possessor a certain number of times and they haven't made it to whatever the equivalent of the end zone is called, they have to kick it and it's now the other team's ball...still, the clock doesn't stop. The only time the game is halted is when a try is scored, when a call is being scrutinized (which doesn't happen often), or someone loses consciousness and/or a limb, and two 40-minute halves with a 10-minute halftime take about and hour and a half to play out. We weren't close enough to see anyone bleeding this time, but it happens - sometimes profusely - and never seems to stop anyone. I can't tell if there are actual plays - I'm sure there are, but I don't know when there is time to communicate them since the game hardly ever stops. It usually goes like this: guy gets ball, passes it behind him or sideways to a teammate (no forward passing) - this happens until the sucker at the end doesn't have anyone left to pass it to and is forced to run dead on into the wall of giant Clydesdale-like men in front of him where they will trample him into the ground. The victim relinquishes the ball to the nearest teammate behind him and they start the "play" again. I don't know all of the differences between Rugby League and Rugby Union, but, while everyone I know seems to prefer Rugby League, I recall loving Rugby Union because of the additional competitive cheerleading-like facet. I don't mean they have cheerleaders on the sidelines - I mean these giant men with thighs the size of tree trunks ban together in a scrum and hoist a teammate high into the air to get the ball before they are chased down the field for their beat down. It really adds to the spectacle in a "picture Paul Bunyan as a competitive gymnast" kind of way.
If you want the particulars, there are no doubt numerous resources that compare these games based on actual fact (if you're into that) - the Guardian has an interesting page that compares stats on players and popularity.