The free product applies to the theory of the fundamental group in topology. If connected spaces X and Y are joined at a single point, the fundamental group of the resulting space will be the free product of the fundamental groups of X and of Y. This is a special case of van Kampen's theorem. The modular group is a free product of cyclic groups of orders 2 and 3, up to a problem with defining it to within index 2. Groups can be shown to have free product structure by means of group actions on trees.
It may not look like an intrinsic definition. The dependence on the choice of presentation can be eliminated by showing that free product is the coproduct in the category of groups.
The more general construction of free product with amalgamation is correspondingly a pushout in the same category.