The Layer by Layer method, also known as the Beginners method is a method of solving the 3x3x3 Rubik's Cube. Many beginners' methods use this approach, and it also forms the basis of the CFOP speedcubing technique.
The main method that people advance to off of beginner methods is CFOP. Nobody has been able to devise a more efficient method and although some different methods are still popular and incredibly fast times are possible on those as well, CFOP has always been the main method.
Roux is a metho that is a large competitor with CFOP. The fastest Roux solver is currently 15th in the World.
Roux uses some pretty advanced techniques such as block building right off the bat, so almost all people who use Roux have switched to it after getting familiar and fairly quick with CFOP.
The Metha method method was made for 3x3 by Yash Mehta, but it has been adapted to big cubes by Matthew Hinton with the Mehta-MH method, and it also works on Square-1
There are multiple versions of Mehta. The main group is Mehta-OS, OS standing for Option Select, and the advanced section is Mehta-TDR.
ZZ method tries to shorten CFOP’s move count and improve finger tricking opportunities. It involves orienting edges as part of the first stage to remove the need for F, B or D moves. The last layer can then be solved in one look as all four edges are oriented, however there are 493 algorithms to learn to do this; something that only a few dozen people have been able to learn in its entirety.