So I understand the mechanics of the fundamental group, but I want to gain a more natural intuition behind it. I imagine the fundamental group $\pi_{1}(X)$ to detect "holes" in a space. For example, it detects the hole in the punctured plane. However, it does not detect all type of holes, specifically the hole at $\mathbb{R}^3-0$. Presumably, $\pi_{2}(\mathbb{R}^3-0)$ could do the job in this case. I was wondering if you guys could give me some more geometrical intuition behind the fundamental group. For example, what types of "holes" can $\pi_{1}(X)$ detect? Let's restrict the discussion to subspaces of $\mathbb{R}^{n}$.
Sorry if this is a repost.