What is the difference between "equality" and "equivalence relation" I think equality is just an instance of equivalence relation. An equivalence relation can be defined in the set theory, but how can we define "equality"? I wonder what equality is. 
 A: In the usual first order approach to set theory, ZFC, your language can be as minimal as just containing the relation $\in$ — equality (=) can then defined through the axiom of extensionality (A = B iff $\forall x (x \in A \iff x \in B)$).
A: Your first sentence: You are right, the relation "equality", namely $a \sim b \iff a = b$ is an equivalence relation: $a = a$ holds for all $a$, $a = b \iff b = a$ and $a = b \land b = c \implies a = c$.
Your last sentence: So, equality can be defined as an equivalence relation on many sets, i.e the integers order real numbers.
A: Let's think about a concrete example.
We can say that two integers are equivalent if their difference is divisible by $2$. Even numbers are then equivalent to each other, without all being equal.
One of the things we can do in mathematics, and which gets done very often, is then to "factor out" by the equivalence relation - to treat elements of equivalence classes as if they are all equal. In the case of even and odd integers we can do this, for example, by saying that we will call the even integers by the name $"0"$ and odd integers by the name $"1"$. Then because odd and even behave well algebraically we end up with a new thing - the integers modulo $2$ with rules like $0+0=0$ replacing "even +even =even".
Sometimes it is very useful to be able to simplify in this way - we can prove things about integers by looking at what happens modulo $2$. It is often useful in elementary questions about square numbers, for example, to work modulo $8$, because every odd integer square belongs to the same class modulo $8$.
