This is the most popular system for speedcubing. Invented by Jessica Fridrich in the 1980s. Click here for my version of it.
Gist: Cross, F2L, OLL, PLL
Invented by Lars Petrus, who placed 4th in the 1st Rubik's Cube World Championships in 1982.
Gist: 2X2X2 block, extend to 2X2X3 block, twist the remaining edges (6 of them), COLL, ELL
A relatively new system invented by Ryan Heise.
Gist: F2L with one slot unsolved, last F2L edge and orient LL edges, Permute edges and 2 corners, Remaining LL
Invented by Ron van Bruchem and Zbigniew Zborowski. Highly advanced LBL system with lots of algs to memorize (too many in fact :P).
Gist: Extended cross, 2 more F2L pairs, insert last F2l pair and flip LL edges simultaneously, LL
Invented by Lars Vandenberg and Dan Harris. A simpler version of the ZB method. Algorithms are a subset of the ZB method.
Gist: Cross, 3 F2L pairs, pair up last F2L pair, insert last F2L pair and flip LL edges simultaneously, COLL, ELL
Invented by the 1st Rubik WC world champion, who set the first world record of 22.95 seconds. This is a corners-first method.
Gist: Orient D-corners, Orient U-corners, Permute Corners, 3 D-edges, 3 U-edges, Complete either the remaining D or U edge, Put in last edge while orienting M-edges, Permute M-edges and centers.
Marc Waterman average 16.4 seconds with this method. This is a highly optimized Corners-First system.
Invented by Gilles Roux. Hybrid of corners first and layer-by-layer method. Relatively new method I would say.
Gist: Build a 1X2X3 block, Build an opposite 1X2X3 block, CLL, Solve remaining edges (6 of them) and 4 M-centers
Invented by David. Flexible method.
Gist: F2L (Place two edge pieces, Place four corners on the first level, Place the other two edges on the middle level, Place the four edges of the first level), CLL+ELL