Here are two examples :
$1 - $ Consider the relation $\equiv$ ( an equivalent relation), then
$$a \sim b \Leftrightarrow a\equiv b \mod 2 $$
That is, $a$ and $b$ will be in the same class $\overline{a}$ if their remainders of the division by $2$ are the same. For example $4$ and $6$ belong to the same class, which we are going to choose a representant $0$, because
$$6 = 3 \cdot 2 + \color{red}{0} \ \ \text{and} \ \ 4 = 2 \cdot 2 + \color{red}{0}$$
then we say $\overline{4} = \overline{6} = \overline{0}$. If we think, there are two distinct classes: $$\overline{0} = \{x \in \mathbb Z ; x \equiv 0 \mod 2, \text{$x$ is even}\}\ \ \text{and}\ \ \overline{1} = \{x \in \mathbb Z ; x \equiv 1 \mod 2, \text{$x$ is odd}\}$$
The set of all classes is
$$\mathbb Z_2 = \{\overline{0}, \overline{1}\}$$
$2-$ Consider the relation
$$(a,b) \sim (c,d) \Leftrightarrow ac = bd $$
This equivalent relation gives us the fractions, that is the filed of fractions of $\mathbb Z$. Similarly we choose a class representant for example,
$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6 } = \cdots$$
we choose $\frac{1}{2}$ to be the class representant. Notice that $\mathbb Q = \{ \frac{a}{b} ; a,b \in \mathbb Z, \ \ \text{where}\ \ b \neq 0\}$ is the set of all classes.
Best Answer
Equivalence classes are sets of elements which are all equivalent between them. For instance if the equivalence relation $\sim$ is "having the same sex", then there are two equivalence classes in the world: boys and girls (if we forget the ambiguous cases). In the same way, if the equivalence relation is "being born the same year", then each year yields a different equivalence class of all the people from this year.
To sum up, an equivalence relation cuts the universe into "potatoes" of elements: inside a potato, all elements are equivalent to each other, and a potato is called an equivalence class.