This is just an extension of Todd's answer to summarize Adamek's example, which is a bit convoluted. By a graph we mean a set equipped with a binary relation, call it ∼. If A is a graph, let $A^{(3)}$ be the set of triples (x,y,z) such that x∼y∼z. And let F(A) be the power set $P(A^{(3)})$, equipped with the relation defined by ∅∼X for all nonempty X. Adamek's category is the category of algebras for the endofunctor F of the category of graphs, i.e. of graphs A equipped with a graph-morphism F(A)→A. He proves that the forgetful functor from this category to Graphs has a left adjoint, namely $A\mapsto A \sqcup F(A)$, and is monadic. But he gives the following example of a pair of parallel morphisms of F-algebras that have no coequalizer.
Let A be the set {p,q} with the empty relation ∼, and let B be the set {s,t} with s∼t only. Then F(A) and F(B) are both the graph {∅} with the empty relation, and we make A and B into F-algebras by sending ∅ to p and s, respectively. Now let f:A→B send p to s and q to t, while g:A→B sends p and q both to s. Adamek goes on to prove that if f and g had a coequalizer in F-algebras, then one could construct from this a weakly initial P-algebra, where P is the powerset endofunctor on Set; from this one could then construct an initial P-algebra, hence a fixed point of P, which contradicts Cantor's diagonal argument.
I don't have time to summarize the construction of the weakly initial P-algebra from a coequalizer of f and g, but I'm making this CW, so anyone else who wants to add it, feel free.
