Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by
$$x^n \mapsto \sum_{k=0}^n {n \choose k} x^k \otimes x^{n-k}$$
describes the ways in which one can decompose a set into two subsets. (Note that the convolution product of linear functionals $k[x] \to k$ can be identified with the product of exponential generating functions, at least in characteristic zero.functions.)
As a more complicated example, there is a coalgebra describing the ways in which one can decompose a connected region of $\mathbb{R}^2$ tiled by finitely many squares into two such regions. There are endless variations on this construction.
I learned this point of view from Gian-Carlo Rota's Coalgebras and bialgebras in combinatorics, which I currently cannot find an online copy of...
