Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It looks like classes are introduced to get away from Russel's paradox how does it?
The Oxford English Dictionary s.v. equivalence quotes Kleene, Introduction to Metamathematics ($1952$), as follows:
Von Neumann $1928$ chooses from each of these sets of sets (‘equivalence classes’ [Ger. Äquivalenzklassen]) a particular set to serve as the cardinal of any set in the class.
This is the earliest citation that clearly refers to the usual modern sense of the term. It’s possible, then, that the English term is simply a calque of the German, and that the question should be why the German term uses Klasse rather than Menge.
The term "equivalence class" is a single term in the mathematical language. It does, however, relate to the term "class" from set theory.
Classes, in modern set theory, are collections which are defined by a formula (perhaps with parameters). An equivalence class is also defined by a formula with parameters. We have two parameters, the equivalence relation and the representative of the equivalence class.
In theories like $\sf ZFC$ every set is a class. Because every set is defined by a formula using itself as a parameter. It's a bit of cheating, but mathematically it is correct. In our case parameters are things we already know are sets, and if $A$ is a set then we can use it as a parameter for the formula $x\in A$.
The difference between proper classes, i.e. classes which are not sets, and sets has been discussed greatly on this site. In a nutshell proper classes are collection we can define (in the language of set theory) but we can prove that they do not form a set. One difference is that while we have a coherent way of assign "size" to a set, we don't have the ability to assign "size" to a proper class.