In Hatcher 1.1.6
We can regard $\pi_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S_1,s_0) \to (X,x_0)$. Let $[S_1,X]$ be the set of homotopy classes of maps $S_1 \to X$, with no conditions on basepoints. Thus there is a natural map $Φ: \pi_1(X,x_0) \to [S_1,X]$ obtained by ignoring basepoints.
I don't quite understand what is mean by "with no condition on base points"(What is mean by "base point" of $S_1$?) and "ignoring basepoints". And whether $[S_1,X]$ is defined to be a set of continuous map or all the maps. What is a formal description of "ignoring basepoints"?
If some one could give an example that given a map in $\pi_1(X,x_0)$, how to get maps in $[S_1,X]$ in the way the author describes. I would very appreciate.