How do we describe the equivalence between $f:(I,\partial I)\to(X,x_0)$ and its "naturally equivalent" map $\tilde f:(S^1, s_0)\to (X,x_0)$ In the theory of fundamental groups and homotopy, we always regard $f:(I,\partial I)\to(X,x_0)$ and $\tilde f:(S^1, s_0)\to (X,x_0)$ as the same thing and change one from another freely. I wonder if there is an advanced terminology in category theory or somewhere that describes their equivalence.
 A: This is what's known as a universal property. For every map $f:(I,\partial{I})\to (X,x_0)$ there exists a unique map $\tilde{f}:(S^1,s_0)\to(X,x_0)$ such that:
$$f = \tilde{f}\circ\pi$$
where $\pi:(I,\partial{I})\to (S^1,s_0)$ is the quotient map. 
Now, we'll set this up categorically. Let $\mathbf{Top}_*$ be the category of pointed topological spaces $(X,x_0)$ with basepoint preserving maps. Define a new category $\mathcal{C}=((I,\partial{I})\downarrow \mathbf{Top}_*)$ whose objects are maps $g:(I,\partial{I})\to(X,x_0)$ satisfying $g(\partial{I})=x_0$ and whose morphisms
$$(g_1(I,\partial{I})\to(X,x_0))\xrightarrow{\ \phi \ }(g_2:(I,\partial{I})\to(Y,y_0))$$
are basepoint preserving maps $\phi:(X,x_0)\to(Y,y_0)$ such that $g_2=\phi\circ g_1$. 
The initial statement I made now reads; for every object $f:(I,\partial{I})\to(X,x_0)$ of $\mathcal{C}$ there exists a unique morphism $\tilde{f}:(S^1,s_0)\to(X,x_0)$ from $\pi:(I,\partial{I})\to(S^1,s_0)$ to $f$. In other words, $\pi$ is an initial object of $\mathcal{C}$. Note also that this is the same thing as saying that precomposition with $\pi$ gives a bijection between the Hom-sets:
$$-\circ\pi:\text{Hom}_{\mathbf{Top}_*}((S^1,s_0),(X,x_0))\to \text{Hom}_{\mathbf{Top^2}}((I,\partial{I}),(X,\{x_0\}))$$
where $\mathbf{Top}^2$ is the category of pairs $(X,A)$ with maps $f:(X,A)\to(Y,B)$ (meaning $f:X\to Y$ is continuous and $f(A)\subseteq B$). So following the wiki page your functor $U$ is the "forgetful" functor $U:\mathbf{Top}_*\to\mathbf{Top}^2$ and your category is $((I,\partial{I})\downarrow U)$.
The same thing holds for quotient groups, rings, etc. I highly suggest reading MacLane's text, where you can find a rigorous treatment of universal properties. 
A: There's an adjunction between $C$, the category of pairs of spaces $(X,Y)$ and $D$, the category of pointed spaces, whose left adjoint is the quotient space construction described in the other answer. The right adjoint is the inclusion of $D$ as a full subcategory of $C$. We say a full subcategory whose inclusion has a left adjoint is"reflective", and that $(S^1,*)$ is the "reflection" of $(I,\delta I)$ in $D$.
