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. 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. 

 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...