Two sets are equal, Definition. Two sets A and B are the same, if 
$ \displaystyle \forall X: [X \in A \Leftrightarrow X \in B] \land [A \in X \Leftrightarrow B \in X]$
Ehm. I know one explanation that says, "two sets are equal, if they own the same elements". If so, why do we need the second part of the definition? If I read this, I see: "Two sets are equal, if they own the same elements and are elements of some other thing". But then, they are elements of that that is an element of themselves. What are these circles of mind?
Can somebody explain me the sacral meaning of this second part? Thanks.

The definition of equality both tells us what equality is but also tells us what facts about equal things we may use. So if we are told $ A = B $ then we are told that both that they have the same elements and we are being told that they are elements of the same sets.
When we are dealing at the level of axioms we can't depend on what we expect equality to mean.  For example imagine if we had symbols $ \doteq $ and $ \triangleright $.  If you had the axiom $ A \doteq B \Leftrightarrow \forall X : [ X \triangleright A \Leftrightarrow X \triangleright B ] $
Then the following graph (with arrows indicating the $ \triangleright $ relationship) would have $ A \doteq B $ but A and B wouldn't act in the way you would expect them to act if they where equal.
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