D&D Adventurers League is an "organized play" system.
Local stores host games (often with volunteer DMs, sometimes with paid ones). Players play point-build or array-build characters in whatever adventure is being run. Character's experience total is tracked, and players can drop-in/drop-out on a session by session basis.
Ideally, it works best if the adventure has the same 3-7 players week after week, but that's not a requirement. In fact, that's often a problem to arrange.
A given character can only participate once in a given adventure. If a player wants to play it again, they can use a different character.
From the DM perspective, the DM gets his marching orders from the store. The store either gives him the download password or gives him the adventure to run. The DM then runs the adventure at the store in the allotted time for the first 7 players who sit down for it.
Wizards keeps track of total player numbers - they do this using the DCI numbers. The store records your DCI number, and reports it to Wizards.
All your XP is tracked on forms that you bring with, not by Wizards.
The store needs to have a coordinator - they keep track of who played, and make certain the DM's have allowed adventures to run.
At present, that's a pretty decent selection. As season 3 is about to begin, all the season 1 and 2 adventures are legal for season 3 play, as is Lost Mine of Phandelver (in the Beginner's Set). So, on day 1 of season 3, there are over 30 adventures the coordinator can choose from.
The coordinators schedule the events - there are certain restrictions on scheduling - and upload the DCI numbers of those who played in those events. They also serve as a safety net - if a player or DM is acting inappropriately, the coordinator can toss them from the event.
For full credit, events are supposed to have their results entered within 24 hours of completion. Note that, for MTG tournies, that includes the winners, but for D&D, it's just who played.
What the Various Elements Mean
Encounters: Play of the "big module" for the season in its reduced form, covering levels 1-4, on Wednesday nights at a FLGS sponsored public game. These modules are coded DDEN, and are released 1 per season. Tends to be about 10-20 weeks of play, depending upon DMing style and player behavior.
Expeditions: play of any of the expeditions modules at an FLGS sponsored public game or at a convention. These modules are coded DDEX, and there are more than a dozen per season. They tend to run 3 to 10 hours of play each.
Casual Play: Any other store or club sponsored public event use of the DDEX and/or DDEN modules, or of the hardcover big adventures, that is being reported to Wizards. Can include store reported home play, and play of the hardcover versions in store.
Epics: convention modules not available to the stores. They look to be set for 4-8 hour play blocks. The modules are coded DDEP, and while generally unavailable, leaks have happened.
Home Play: a limited exception to the program, certain modules can be played at home without reporting. It is only allowed for the hardcover modules (At present, Hoard of the Dragon Queen, Rise of Tiamat, Princes of the Apocalypse, and Rage of Demons) and the Lost Mine of Phandelver boxed set.
Note: Some of these definitions are taken from non-public sources, namely the instructions to stores on how to report events. The rest are from the Season 3 Player's Guide.
Benefits to Store
Wizards cuts stores discounts based upon total DCI number using events - Magic Tourneys and Open Play events, as well as D&D Encounters, Expeditions, and Casual Play events.
They also expect that Encounters brings people into the stores. Further, some casual players will show up to play, and decide to buy dice, minis, snacks, or even rulebooks.
Benefits to Players
You can almost always find a D&D Encounters or Expeditions game to play in.
You get to play with people before you decide to invite them to your home game.
The adventures are in fact pretty good.
Because there are numerous pregen characters, plus the option to use characters built by the basic rules (which are free online), it's a chance to have new players find the game by showing up and learning to play at the FLGS.
There also is the opportunity for community awareness. Having D&D played in public really helps reduce the stigma of D&D. When people see it is something that can be played in public, not just in the basement, it helps dispel a lot of the myths.