AA has three elite levels, Gold (lowest), Platinum, and Executive Platinum. Gold and Platinum earn electronic upgrades (called "stickers" because they used to be paper stickers which you would paste to a paper ticket) by flying. Each sticker will allow you to upgrade 500 miles (or less) of flying, so a
flight of 1235 miles would require 3 stickers (so would a flight of 1550 miles, as they give you 50 miles leeway). Gold and Platinum members earn 4 stickers for each 10000 miles that they fly. Executive Platinum members neither earn nor use stickers - they get all their flights upgraded (space available) for free. Exec. Plat. upgrades can clear at 100 hours before the flight, Plat. at 72 hours, and Gold at 24 hours, so the higher your level, the better your chances.

AA does not normally comp elite status (unless your employer's travel department can arrange it for you - it's not something you can do for yourself). You can earn elite status on either miles or points. Each flight will earn both. Mile earned equal miles flown (but 500 minimum per flight). Points are earned as a multiple of miles depending on fare. First, Business, and full-fare coach earn 1.5 points per mile flown, discount coach 1.0, and deep discount coach 0.5, and again a minimum of 500 points per flight. If you earn either 25,000 points or miles in a year you qualify for Gold, 50,000 = Platinum, and 100,000 = EXP.

Instead of status comps, AA offers "challenges" that are a fast track to status. Challenges operate on points only. 5,000 points in any 90 day period will get you Gold, and 10,000 points will get you Platinum. There is no challenge for Executive Platinum - you have to get that the "old fashioned way" - butt in seat.

This is, of course, just an overview. If you have any specific questions, I would be glad to try to answer them.