What would the fundamental group of disjoint union look like? Fundamental group of a wedge sum is a free product of fundamental groups. Hence, $\pi_1$ maps a coproduct of topological spaces with base points to a coproduct of groups.
Since disjoint union(coproduct of topological spaces) is somewhat more free than wedge sum(coproduct of topological spaces with base points), I'm curious what would the fundamental group of disjoint union be. I cannot think of a somewhat product of groups that is more free than the free product.
To be concrete, let $X_i$ be a family of topological spaces. What is $\pi_1(\coprod X_i)$? Can it be expressed by $\pi_1(X_i)$'s?
 A: As mentioned in the comments, there is no such thing as the fundamental group of a space. What there is is the fundamental group of a space $X$ at a basepoint $x$, which is "unique up to isomorphism" if $X$ is path-connected. But a nontrivial disjoint union of spaces is never path-connected, so you really need to pick a basepoint. If you pick a basepoint $x \in X_i$, the fundamental group will just be $\pi_1(X_i, x)$.
This is a simple example of what fundamental groupoids can do for you. Unlike the fundamental group, the fundamental groupoid does not require a choice of basepoint. The fundamental groupoid of a disjoint union is just the disjoint union of fundamental groupoids. 
This fussing with basepoints may seem like pedantry but it becomes quite important once you want to discuss any extra structure. For example, suppose a group $G$ acts on $X$. Does it act on "the" fundamental group? The answer is not necessarily, not even after choosing a basepoint, because $G$ may not preserve any point of $X$! But $G$ always acts on the fundamental groupoid. 
A: When I thought of using fundamental groupoids in van Kampen type situations in 1964-5, I thought it was great to get rid of base points. But then I realised that to compute the fundamental group of the circle I had to reduce to $2$ base points. So my published paper in 1967 used the fundamental group $\pi_1(X,S)$ on a set $S$ of base points, and this idea was developed in the book published in 1968, and now available as Topology and Groupoids (T&G). This is the only topology text in English to use this notion (why is that?). So it is,  as Peter May described the 1988,  differently titled,  edition, "idiosyncratic". However this paper on ODEs  has an application of the notion! 
Here is an example of a connected union of 3 non connected spaces:
 
Note that the fundamental groupoid on the shown set of base points will still capture the symmetry of the situation. Methods of computing colimits of groupoids are analogous to those for groups except that one needs the additional construction of a new groupoid $f_*(G)$ on the set $Y$ from a groupoid $G$ and a function $f:Ob(G) \to Y$. See T&G and Philip Higgins 1971 book Categories and Groupoids (downloadable) (these write $f_*(G)$ as $U_f(G)$). This construction  corresponds to identifying some points of a set of base points, i.e. identifications in dimension $0$. This can't be seen with just one base point, and has useful analogues in higher dimensions. 
To come nearer to the question, and to Qiaochu's answer, for a disjoint union of connected spaces, you can choose one base point in each component.  
So I feel the fundamental groupoid itself is often too big, and the fundamental group is very often too small, but the fundamental groupoid on a set of base points can be chosen to be just right. 
To deal with actions of a group on $X$ you need to take $S$ as a union of orbits of the action. This is developed in Chapter 11 of T&G. 
