Definition of the fundamental group Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What would go wrong if it was defined so? Or is it simply not useful?
 A: We need the homotopy equivalence relation to capture the "essential" parts of the topological space. You can think of it kind like throwing lassos into the space and pulling tightly. 
Without the equivalence relation, we fail to get an actual group. We can't even have an identity element, for instance. 
To expand: the only obvious candidate for an identity element is the constant map from $[0,1]$ to the basepoint (the "standing around, twiddling one's thumbs" path). If you concatenate this with any other distinct path $\gamma$, then you will not end up with $\gamma$. It will have the same image, to be sure, but it's not just the journey, it's how you get there (to abuse an english phrase).
A: You can use paths or loops of variable "length" to get an associative structure with identities, but you do not get strict inverses.  One precise definition  is that a path of length $r$ in a space $X$ is a map $a: [0,r] \to X$ and this composes with $b:[0,s] \to X$ to give $a+b: [0,r+s] \to X$ if and only if $a(r)=b(0)$. Others, for good reason, prefer to define a path to be a pair $(a,r)$ where $r \geqslant 0$ and $a: [0, \infty) \to X$ is constant on $[r,\infty)$. These are called "Moore paths".  
Interestingly, in higher dimensions there is a  tendency to talk about the "fundamental $\infty$-groupoid of a space $X$, but this again is not a strict structure, and so, strictly speaking, does not generalise the fundamental groupoid. The higher structures referred to here do generalise the fundamental groupoid, but are defined only for filtered spaces, which turns out not to be a disadvantage. 
Later: A question is: why take equivalence classes? As mentioned by Simon, we need more manageable invariants.  So there are Seifert-van Kampen Theorems for the fundamental groupoid and group, which allow explicit and useful calculations, given in many books. Analogously, the strict higher groupoids also lead to explicit calculations. 
A: This question brings to mind a definition of the fundamental group of a based space $(X, x_0)$ which may allow you to connect the two concepts. Start with the $\textit{loop space}$ of $X$, which consists of all the continuous pointed maps from $S^1 \subset \mathbb{C}$ to $(X, x_0)$:
$$ \Omega(X, x_0) := \{ f: (S^1, 1) \xrightarrow{cts} (X, x_0) \} $$
This space can be given a topology (the $\textit{compact-open topology}$) in such a way that a path $I \to \Omega(X, x_0)$ corresponds precisely to a continuous map
$$ I \times S^1 \to X,\ (t, s) \mapsto \gamma_t(s) $$
such that each $\gamma_t$ is a pointed map from $S^1$ to $X$, i.e. an element of $\Omega(X, x_0)$ (one fancy way to get this equivalence is to  notice that is that there is an adjunction of functors lurking in there). 
Now, one may naturally be interested in talking about the connected components of this loop space. If we write $\pi_0(Y, y_0)$ to denote the set of path-components of a based space $(Y, y_0)$, with the path component containing $y_0$ singled out, we may $\textit{define}$ the fundamental group of $(X, x_0)$ to be:  
$$ \pi_1(X, x_0) := \pi_0(\Omega(X, x_0), c_{x_0}),$$
where $c_{x_0}$ is the constant loop at $x_0$.
Hence homotopy of paths comes up as a formulation of what it means to connect two elements of the loop space of $(X, x_0)$. So in a sense, if one believes that connectedness is a natural, then it is possible to think of the fundamental group as measuring the connectedness the loop space of $X$. 
A further advantage of this viewpoint on $\pi_1$ is that you can then naturally define the $\textit{higher homotopy groups}$ of $(X, x_0)$ in an iterative fashion, by setting
$$ \pi_{k+1}(X, x_0) := \pi_k(\Omega(X, x_0), c_{x_0}).$$
(The above is inspired from the introduction to Bott and Tu's $\textit{Differential Forms in Algebraic Topology}$.)
A: For a start, it wouldn't be a group because there are no inverses (that is if you use composition of paths as the operation, I am not sure what you mean by product). 
Also the structure isn't really useful, I suppose. For example your structure would be in general uncountably generated.
