Let's start from the end. The best player who plays "false 9" today is, without any doubt, Lionel Messi.
There isn't one definition of a "false 9", but I will try to explain it from my understanding.
"False 9" is generally a player who isn't a pure striker as defined in the past. This player will drop deep into the field to be a part of or start the attack. When doing that, opponents / defenders will follow him and leave more space to players from the 2nd line to threaten the goal.
The "false 9" usually has great finishing, a very high level of vision on the pitch, and good dribbling and passing.
Zonal Marking defines it as:
A unconventional lone striker, who drops deep into midfield. Francesco Totti perhaps invented it for Roma in 2006/7, Lionel Messi played here when he swapped positions with Samuel Eto’o for Barcelona in 2008/09, and Robin van Persie played the role for Arsenal at the start of the 2009/10 season.
Teams who play with a "false 9" generally play in 4-6-0 formation with the "false 9" turning into a striker when needed. I answered a similar question about this formation here: Has 4-6-0 ever worked in high level football?
Another great explanation of the "false 9" according to Barcelona and Josep Guardiola's dream about a team with a goalkeeper and 10 midfielders can be found at the following post.
