What's the meaning of $\succ$ operator? What is the meaning of $\succ $ symbol? 
I have an snippet which includes this operator: (Article is about choice theory)

"a complete and transitive preference $\succ$ over X" 

(Maybe very preliminary question but intrestingly ther is no explanation in the internet about this symbol)
 A: Here is how you would find an explanation on the internet. 
Entering the keywords "complete transitive preference choice theory," the first hit is this wikipedia page on preference. In the section basic premises, they start to define symbols for preference, and then in this later section they define what a complete and transitive preference order is.
I have a slight feeling you might be too focused on the symbol $\succ$ itself. The wording suggests you might believe that the symbol has one and only one meaning. This isn't a very practical viewpoint since many symbols have uses in different concepts, and sometimes the same concept uses multiple symbols. (In case my feeling completely wrong, I apologize in advance!)
Aside from the symbol $\succ$, an author could choose to use $>$ or anything else shaped like that. So the phrase

"a complete and transitive preference $\succ$ over X" 

already completely explains what $\succ$ is: it is a complete and transitive preference over $X$. If one is still puzzled at this point, then the right question is what is a complete and transitive preference order? not "What's this symbol?" The symbol isn't that important, but what it denotes is important.
A: $\succ$ is named "succeeds" and is the name given by the author to the complete and transitive preference (~order). You can read it as a $>$ for simplification.
Other uses of the symbol occur for example with matrices. $A\succ 0$ means that $A$ is a positive definite matrix (i.e. $x^T A x > 0 \quad\forall\ x\neq 0$)
