This paper is dealing with entire solutions of a bistable reaction-diffusion equation with Nagumo type nonlinearity, so called the Allen--Cahn equation. Here the entire solutions are meant by the solutions defined for all $(x,t)\in \mathbb{R}\times\mathbb{R}$. In this article we first show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of $x$-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. We also show the existence of an entire solution emanating from the unstable standing pulse solution and converges to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle we prove a rather general result on the existence of an unstable set of an unstable equilibrium to apply to the present case.