What is the index of a subgroup $H$ in a group $G$? I have read multiple definitions so far but something is not clicking.
My most naive understanding is that $|G:H|$ is a "number" (could be infinite) that represents how many times $H$ is in $G$.
But even this doesn't seem fully correct.
I would like a general non-formal explanation and perhaps an example to understand the intuition.
 A: Upgrading my comment to an answer:
You have the right intuition, but I would word it slightly differently.
$|G:H|$ is the number of (left or right) cosets of $H$. A coset is not a copy of $H$; indeed, a coset of $H$ isn't even a subgroup unless it's $H$ itself.
It would be more correct to call the coset $aH$ a translate of $H$ by the element $a$. Distinct cosets are disjoint and form a partition of $G$. So one might say that the cosets of $H$ constitute a "tiling" of $G$, and $|G:H|$ is the number of tiles.
To take a concrete example, let $G$ be the additive group of integers modulo $12$, and let $H$ be the subgroup generated by $4$, so $H = \{0, 4, 8\}$. Then there are four distinct cosets of $H$, namely:
$$\begin{aligned}
H &= \{0, 4, 8\} \\
1+H &= \{1, 5, 9\} \\
2+H &= \{2, 6, 10\} \\
3+H &= \{3, 7, 11\} \\
\end{aligned}$$
In other words, the cosets are $H$ and its translates $1+H$, $2+H$, and $3+H$.
Together these four cosets contain all the elements of $G$, so they constitute a partition (tiling) of $G$. The number of tiles is the number of cosets, which is $|G:H| = |G|/|H| = 12/3 = 4$.
A: Your naive understanding is roughly correct, especially for the finite groups.
Here are two examples in the infinite case in the same setup (you should draw pictures while reading this).
The set of all nonzero complex numbers, is a group under multiplication of complex numbers, call it $G$. This can be pictured as the plane with a hole at the origin.  Let $H$ be the subset there consisting of number of modulus 1 (unit circle). One checks easily that $H$ is a subgroup.
It is clear to visualize that all concentric circles centred at the  origin one each for every positive number as the radius, cover the whole of  $G$.
Any particular circle of radius $R$, each point there is obtained by multiplying elements of $H$ (the unit circle) by $R$.
 Each of this circle is a coset and they are all disjoint, with the whole group as their union. The index here is number of possible values for $R$ (which is the set of positive real numbers), the number of cosets.
Now coming the other way around, now take $K$ to be the set of positive real numbers as a subset of $G$. This $K$ is also a subgroup of $G$, geometrically this is a ray.
Now rotate this ray by various angles around the origin. For each angle you get a different ray, always starting from the origin. These rays also cover up the group $G$, and they are disjoint. One each for every possible angle in the  semi-closed interval $[0, 2\pi)$.
Each such ray is obtained by multiplying every number in $H$ by the fixed complex number $e^{i\theta}$, so they are all cosets of $K$.
To summarise: in the first case the circle is a subgroup  and the index is infinite with one coset corresponding to every possible positive number as radius.
In the second case positive real numbers form a subgroup with index again infinite, corresponding to every possible angle.
NOTE: When you study the concept quotient groups the above example is worth revisiting and one can see that this says $G/H$ is $K$ and $G/K$ is $H$, or that $G$ is the direct product of $H$ and $K$. (Here we are working with abelian groups and  hence normality is automatic for all subgroups).
