Let's start from the end, The best player who plays "False 9" today is, with out any doubt, Leonal Messi.
There isn't one definition to a "false 9" but I will try to clarfy it as for my understanding.
"False 9" is generally a player who isn't a real pure striker as defined at the past, this player will drop deep into the field to be a part or start the attack.
When doing that oppenents defenders will follow him and leave more space to players from 2nd line to threaten the goal.
The "False 9" usally has, exapt of great Finishing, very high level of Vision on the pitch, Dribelling and Passing.
Zonal Marking define 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 played with "False 9" generally plays in 4-6-0 formation when the "false 9" turning into a striker when needed. I answered simillar question about this formation that you read at the following question - 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.