Confusion on understanding a proposition on equivalence classes I am given to prove this proposition on equivalence classes.

Each element of $A$ is an element of one and only one equivalent class.

The part that is confusing is one and only one. It sounds like an existential statement, the existence of an equivalence class and that that equivalence class is unique. How should one understand this proposition in "common" logic terms, namely "for every", "there exists", if..then,...?
So far here is my understanding, or say 'translation' of the proposition:

There exists a unique equivalence class such that each element of $A$ belongs to that equivalence class.

This understanding is doubtful, at least for me, because it implies that all elements of $A$ belong to one equivalence class, which is wrong. 
I am not looking for a proof, just an understanding of the proposition. Thanks!
 A: If $E$ is an equivalence relation on $A$ then for every $a\in A$ there exists an equivalence class $B$ such that $a\in B$, and for every equivalence class $C$, $a\in C$ implies $C=B$.
So there exists such equivalence class, and it is unique.
Note, however, that $\forall a\exists B\ldots$ is not the same as $\exists B\forall a\ldots$. The former is what I wrote above, and the latter is the statement that all the elements are in the same equivalence class.
A: 

There exists a unique equivalence class such that each element of A belongs to that equivalence class.

This understanding is doubtful, at least for me, because it implies that all elements of A belong to one equivalence class, which is wrong.

You are right to be doubtful, because your translation is wrong.  You are being asked to prove that for each element $a$ in $A$, the element $a$ belongs to exactly one equivalence class.  But your translation, as you observed, says there is one equivalence class to which every element of $a$ belongs.
This subtle difference catches many beginning mathematics students.  But the order of the quantifiers ("there exists" and "for each") is important.  For example, it is true that for each number $x$, there is exactly one number $y$ that is the square of $x$.  But it is completely false that there is exactly one number $y$ which, for each number $x$, is the square of $x$. 
It is correct to say that a number $x$ has one and only one square.  But we don't mean by this that every number has the same square!  And similarly when we say that an element of $A$ is in "one and only one" equivalence class,  we don't mean that every element is in the same class.  We just mean that each element is in one class, not zero, and not more than one.
A: You are asked to show that any given element $x\in A$ belongs to precisely one equivalence class. Thus, that for that given element $x\in A$ there exists an equivalence class $E$ such that $x\in E$ and that if $F$ is any other equivalence class with $x\in F$ then $E=F$. Of course for any other element $y$ its unique equivalence class might be different than $E$.
