For any path-connected space $X$, the set $\pi_n(X)$ is the set of homotopy equivalence classes of continuous functions $f:S^n \to X$ (some authors use loops instead). $\pi_n(X)$ is also a group under the operation $*$. Thus, we can discuss isomorphisms between $\pi_n(X)$ and other groups. For instance, $\pi_1(S^1) \cong \mathbb{Z}$. I recognize that some authors (rather lazily in my opinion) will right this as $\pi_1(S^1) = \mathbb{Z}$. However, what does the notation $\pi_n(X) = 0$ and $\pi_n(X) = 1$ mean?
Edit: For some clarification for where I pulled these notations out of, I got them from these two statements from my topology class:
If $X$ is a contractible space, then $\pi_n(X) = 0$ for all $n \ge 0$.
A topological space $X$ is called simply connected if it is path connected and $\pi_1(X) = 1$.