Regarding the category of topological spaces with arrows being homotopy classes of continuous maps This is an exercise from Leinster's book:

There is a category $\textbf {Toph}$ whose objects are topological spaces and whose maps $X\to Y$ are homotopy classes of continuous maps from $X$ to $Y$. What do you need to know about homotopy in order to prove that $\textbf {Toph}$ is a category? What does it mean, in purely topological terms, for two objects of $\textbf {Toph}$ to be isomorphic?

Even though they are explicitly described, I don't understand what the arrows are. Say, in the category of topological spaces with continuous maps, there is an arrow $A\to B$ iff there is a continuous map $A\to B$. But similar description doesn't work for our case because homotopy classes of continuous maps are not maps. How should I think of arrows in this case?
 A: In the usual category of topological spaces it is incorrect to say "there is an arrow $A\to B$ iff there is a continuous map $A\to B$". Instead, the arrows in the category are the continuous functions. There is precisely one arrow $A\to B$ for each continuous function $A\to B$. More clearly: $\mathbf{Top}(A,B)=\{f\colon A\to B\mid f \mathrm{\ is \ continuous}\}$.
Similarly then, each arrow in $\mathbf {Toph}$ is an equivalence class of continuous functions where the equivalence relation is that of being homotopic. It's not a particularly easy to imagine category. 
Remark: Your confusion may stem for the terribly incorrect assumption that arrows in a category must somehow be functions. That is not the case. The objects and the arrows in a category are abstract entities. They need not be sets and functions at all. 
A: First, lets go over the definition of what a category $C$ $is$.
$C$ consists of a collection of objects $ob(C)$, this can be a collection of any objects you would like. It can be the collection of all fruits, natural numbers or names of animals, (note that $ob(C)$ is not required to be a set). C also consists of a $rule$ assigning each two objects X and Y to a set $Hom_C(X, Y)$ (could be empty) of arrows $X \rightarrow Y$, we say that $f$ is an arrow $X \rightarrow Y$ if $f \in Hom_C(X,Y)$. 
This can be a set of whatever you would like, even things not classically thought of "maps", if $ob(C)$ is the collection of all animals maybe $Hom_C(X, Y)$ is the set of diseases that may be transfered from animal X to animal Y or something equally stupid. You get the point.
$C$ also consists of a function $\circ_C :Hom_C(Y, Z) \times Hom(X,Y) \rightarrow Hom_C(X,Z)$ for every three objects $X$ $Y$ and $Z$. This function is called composition.
Composition of arrows is also required to be associative and an identity arrow for each object is also required to exist. This is an arrow $id_X:X \rightarrow X$ such that $f \circ id_X = f = id_Y \circ f$ for all $f: X \rightarrow Y$ and for all $X$ and $Y$.
Finally we see that if we define ob(C) to be the class of topological spaces with $Hom_C(X, Y) =$ {homotopy classes of maps $X \rightarrow Y$} and composition to be defined as $[f]\circ_{C} [g] = [f \circ g]$ where $\circ$ is regular composition of continuous maps. 
Identity is obvious, associativity is aswell. The real challenge is to show that composition is even well defined. What I mean by this is that if $[f_1]=[f_2]$ then $[f_1 \circ h] = [f_2 \circ h]$ and $[g \circ f_1] = [g \circ f_2]$ for $g$ and $h$ such that the expressions are defined. First of all, try to understand what $[r]=[t]$ really means in topological terms.
A: The other answers definitely explain the concept of homotopy classes of continuous maps. I just wanted to add that you maybe already worked with that notion. One definition for the fundamental group $\pi_1(X,x)$ of a pointed topological space $(X,x)$ is given by the homotopy classes of continuous maps $[0,1] \rightarrow X$ that map the boundary (the points $0$ and $1$) to the basepoint $x \in X$. Thus the elements of $\pi_1(X,x)$ are homotopy classes of loops at the point $x$, which means that every element $g \in \pi_1(X,x)$ is a loop up to continuous deformations. I have drawn a picture to visualize that:

The red and the blue loop can be deformed contiuously into each other and thus yield the same element in the fundamental group, while the green loop cannot be shrunken to the basepoint because of the hole and thus is not the identityy element.
