Let $X$ be a topological space and $A \subset X$ a subspace. For $x \in A$, the homotopy group $\pi_n(X,A,x)$ of the pair $(X,A)$ is by definition the set of homotopy classes of n-cells relative to $X$ modulo $A$ (the usual definition).
Question1: Is $\pi_1(X,A,x_0)$ always trivial?
Question2: If A = X, $\pi_n$ is trivial for all $n$? How?
If the image of $I^n$ lies entirely in $A$, then you can contract it, since $I^n$ is a contractible space. Since you added a question, $\pi_1(X,A)$ is homotopy classes of maps $I\rightarrow X$ so that the endpoints of $I$ are contained in $A$.