In $\ZF$, everything is a set. Nevertheless, we introduce classes to improve our quality of life. A class is an informal, set-like object that describes a collection of sets satisfying a certain property. Through classes, we can treat formulas like sets.
Classes only play a role like syntactic sugar. Arguments involving classes are ultimately reduced to formulas.
If $\varphi(x,p)$ is a formula with free variables among $x$ and $p$, then we define the class
\[C = \{x:\varphi(x,p)\}\]as an object such that
\[\forall x :\enspace x\in C \enspace\lrimp\enspace \varphi(x,p).\]Relations and operations on classes, such as
\[C=D ,\enspace C\subseteq D ,\enspace C\cup D ,\enspace C\cap D ,\enspace C\setminus D ,\enspace \bigcup C ,\enspace \bigcap C,\]are defined similarly to those on sets.
$C$ is a subclass of $D$ if $C\subseteq D$.
If a class $C$ is not a set, i.e. there is no set $X$ such that
\[\forall x :\enspace x\in C \enspace\lrimp\enspace x\in X,\]$C$ is called a proper class.