Notational Definition: $A \subset B$ Suppose $A$ and $B$ are two sets and suppose $A \subset B$, then from the definition of subset, it implicitly follows that $A$ might also be equal to $B$.
Then why do we need $A \subseteq B$ where we explicitly declare that $A$ might also be equal to $B$?
Does $A \subset B$ means $A$ is a proper subset of $B$?
 A: Some text books define $\subset$ as $A\subseteq B$. Others define it as $A \subsetneq B$. I think it depends on the particular book/author you are reading.
A: This is a particular place where notation gets confusing and inconsistent. It seems like the most common interpretation of 
$$A\subset B$$
is that $A$ is a proper subset of $B$. Sometimes, however, it can be interpreted as subset but possibly equal to:
$$A\subseteq B.$$
Some authors also prefer 
$$ A\subsetneq B$$
with the intention of specifically stating that $A$ is not equal to the containing set. You should consult the book's definition or ask your instructor what $A\subset B$ is defined as, but more often than not, it denotes proper subsets.
A: This is a just a conventional thing. Different people prefer different things. I've seen some texts make $A \subset B$ include the possibility that $A = B$ and some that make $A \subset B$ mean that $A$ is a proper subset of $B$. I recommend using $A \subseteq B$ when $A = B$ is a possibility and $A \subset B$ when $A = B$ is not a possibility.
