BitTorrent: BitTorrent is a protocol designed for transferring files. It is peer-to-peer in nature, as users connect to each other directly to send and receive portions of the file. There is also one central server (called a tracker) which coordinates the action of all such peers.

Terminology: Here's some basic terminology you'll want to get firmiliar with:
Torrent - The file which tells your client what to download. Essentially it's the key to your download and your client is the door.
Tracker - A server on the Internet that acts to coordinate the action of BitTorrent clients.
Client - This can also be referred to as your program. It is the service you use to download your programs. (i.e. µTorrent, Azureus, etc.)
Peer - Another computer connected to the peer to peer (p2p) network to share information.
Peer to Peer (p2p) - A term used to describe file sharing from computer to computer rather than downloading from a file server.
Direct Download - A download from a file server to a computer.
Seeders - Those who let others download files from them, essentially sharing their bandwith and file with a group of peers.
Leechers - Those who download the files. Sometimes Leechers download a torrent file and close the BitTorrent client right afterwards; this is very rude and very frowned upon by all involved in the BitTorrent community.
Reseeding - This is a phenomena that occurs when no one is seeding a specific torrent. Reseeding is the act of putting a file back onto the torrent client for others to download upon request (Usually the requester will seed the file for a lengthened period of time as gratitude for the reseeding.)
[Batch] - This is a term you usually see at the end of a torrent filename. This is used to describe a torrent that allows you to download multiple files into the same folder (i.e. ADC_One_Piece_220_230_[Batch].torrent) Usually if you only want one specific file from the batch and cannot find an individual torrent for it then your client will allow you to pick that specific file from the [Batch] torrent.

Why Use BitTorrent: There are many different reasons why fansubbing groups such as ADC, KF, Dattebayo, and Gerusama use BitTorrent as their preferred method of distributing their releases, here are a few:
It's Cheaper - Using BitTorrent frees up Bandwith for the Fansubbers' Websites by reallocating it between seeders and leechers.
Files become available longer - As long as people are seeding the files on a BitTorrent client then the download is still available.
It's actually faster* - Because bandwith is switched from one server to many different peers dealing directly with one another more bandwith becomes available and leechers do not consume all the filer server's bandwith.