Speedcubing methods are a hotly debated topic. In this blog post, I'll try to keep my answers as objective as possible. Keep in mind that these are opinions, and you are free to disagree.
The fastest methods for two-handed solving are CFOP and Roux. This might make some people mad right off the bat, but here is my reasoning.
About 95% of top cubers use CFOP, and the remaining cubers use Roux. Now some people might use this is evidence that CFOP is better than Roux, but let's not jump to conclusions.
CFOP is a lot more popular than Roux for two reasons. First, it was invented much earlier. CFOP was invented in 1981, whereas Roux was invented in 2003. That's over 20 years for CFOP to get a head start. Second, CFOP is more similar to the method that beginners learn, layer by layer. CFOP starts with the cross, just like LBL. It ends by solving the last layer, similar to LBL, but with fewer moves.
In all likelihood, CFOP will probably remain the most popular method for a long time, if not forever. But I don't accept that as evidence of CFOP being better than Roux.
CFOP and Roux both have pros and cons. CFOP solvers can achieve higher turns per second (TPS) because the last layer is extremely ergonomic. On the other hand, Roux uses less moves, so high TPS is not as important.
Many cubers agree that Roux is the best method for one-handed solving. This might still be pretty controversial though.
Roux is a controversial method for one-handed because Rouxers use the table heavily. Some people call this table abuse.
While solving the last six edges (LSE), a Rouxer rests the cube on the table. LSE only uses the M slice and the U layer. This allows Rouxers to turn with remarkable speed during LSE.
In addition, Roux's low movecount is a big plus for one-handed solving. It's harder to get high TPS with one hand, so move efficiency gives Roux an edge over CFOP.
Keep in mind, many people do well with CFOP for one-handed solving. But if you want to specialize in one-handed solving, it will make your life easier to use Roux.
CFOP is undoubtedly the fastest method for big cubes. This is because the Yau method for big cubes works very well with CFOP.
You can do something similar to Yau with Roux, and that method is called Meyer. However, Meyer is still not as good as Yau for a couple reasons. Yau has more freedom during the edge pairing stage, and M slice moves are awkward on big cubes.
Technically, Rouxers can be fast at big cubes, but it takes a little more work. Like before, I recommend big cube specialists to use CFOP.
Layer by layer (LBL) is the easiest method to learn. It's the method that nearly every beginner learns.
LBL is so easy because it breaks down into a series of small steps. It also only requires about 4 algorithms, depending on which variant you learn.
Since LBL is the easiest method, it is also one of the slowest. For cubers who want to take it to the next level, it is recommended to learn a more advanced method that uses less moves.