The definition I found on a book is
If (A, R) is a poset, then an element x ∈ A is called a maximal element of A
if for all a ∈ A, a != x => x !R a.
The contrapositive statement is
x is a maximal element if for each a ∈ A, x R a => x = a.
If it apply to >= relation on A = {1,2} where R = {(2,1),(2,2),(1,1)} is a poset and based on this definition 1 is maximal element. Is this correct? Am I missing something?
Yes, but the book uses the (usual) convention that $(a, b) \in \mathcal R$ (or, with another notation, $a \mathcal R b$) should be understood as "$a$ is smaller than $b$". Of course, it does not really matter, but the definition of "maximal element" that you use makes more sense if you understand the order in this direction.
Now, if we want to reformulate the definition of the book, the second one says that $x$ is a maximal element if for each $a \in A$, $x \leq a \Rightarrow x = a$, meaning that we cannot find an element $a$ greater than $x$ - which is exactly the definition of a maximal element.