A reaction-diffusion initial-boundary problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is discussed; it is shown that in general the solution has a weak singularity near the initial time $t=0$, and sharp pointwise bounds on the derivatives of this solution are derived. These bounds are used in a new analysis of the standard L1 finite difference method for the time derivative combined with a standard finite difference approximation for the spatial derivative. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading to solve the problem numerically.