Question: What does the notation $[G : H]$ mean in group theory? Does it mean the same as quotient group? I have seen the above notation a lot while studying group theory but it still remains unclear to me about the meaning of this notation?

Answer: 
The index $[G:H]$ is the number of left cosets of $H$ in $G$ and it is the number of right cosets of $H$ in $G$. That those numbers are equal is obvious when $G$ is abelian, but it is true for all groups and this does *not* require $G$ to be finite, so the equality is not fully explained by relying on Lagrange’s theorem for finite groups.

For each left $H$-coset $gH = \{gh : h \in H\}$, inverting its elements gives us
$$\{h^{-1}g^{-1} : h \in H\} = \{hg^{-1} : h \in H\} = Hg^{-1},
$$
which is a right $H$-coset, and sending each $gH$ to $Hg^{-1}$ is a well-defined bijection from the left $H$-cosets to the right $H$-cosets, even if the number of cosets is infinite.

For example, $[\mathbf Z: m\mathbf Z] = m$ for positive integers $m$, $[\mathbf R^\times:\{\pm 1\}]$ is infinite, the positive reals $\mathbf R\_{>0}$ have index $2$ in $\mathbf R^\times$, and the group of real $2\times 2$ matrices with positive determinant ${\rm GL}\_2^+(\mathbf R)$ has index $2$ in the group of real invertible $2\times 2$ matrices ${\rm GL}\_2(\mathbf R)$. That a subgroup of index $2$ is a normal subgroup holds for infinite groups, not just finite groups.

The index, when infinite, should properly be considered as a cardinal number, but typically in a first group theory course you may do very little with subgroup indices when they are infinite, so you might just lump all cases of infinite index together as “$[G:H]=\infty$” in the same way infinite sets $S$ are all lumped together as “$\# S = \infty$” unless there is a good reason to use the more careful notion of cardinal numbers.
We understand mathematics through examples, we appropriate notions through examples that we construct.
Let's take an example that is simple enough to be understood by everyone.

Let $G:=(\mathbb Z/6\mathbb Z,+)=(\{\bar0,\bar1,\bar2,\bar3,\bar4,\bar5,\},+)$ and $H:=\{\bar0,\bar3\}$.

$G/H=\{H,\bar1+H,\bar2+H\}=\{\{\bar0,\bar3\},\{\bar1,\bar4\},\{\bar2,\bar5\}\}$.

Each coset has two elements, the same number of elements as $H$. We also know that $G/H$ is a partition of $G$. So the number of elements of $G$ is equal to the number of cosets, which we denote by $$[G:H]$$(here, $[G:H]=3$).

This notion is interesting and deserves to be introduced: for example, we deduce from the above reasoning easily extendable to the general case that$$|G|=|H|\times[G:H]$$
In particular, for any subgroup of a given group, the order of the subgroup divides that of the group.
Here, 2|6.