We investigated the wetting characteristics such as contact angle, wetting radius and topography of water droplets on smooth and random solid surfaces. Molecular dynamic simulation is employed to analyze the wetting behavior of water droplets on smooth and rough surfaces by considering different potential energy models of bond, angle, Lennard-Jones and Coulomb to calculate the interacting forces between water molecules. The Lennard-Jones potential energy model is adopted as an interaction model between water molecules and solid surface atoms. The randomly rough surface is generated by changing the standard deviation of roughness height from $1;{AA}$ to $3;{AA}$ with the fixed autocorrelation length. The size of water droplet considered is in the range from 2,000 to 5,000 molecules. The contact angles increase generally with increasing number of water molecules. For a hydrophobic surface whose characteristic energy is 0.1 kcal/mol, the contact angles depend rarely on the standard deviation of the roughness height. However, when the surface energy is 0.5 and 1.0 kcal/mol, the contact angles depend on both the roughness height of surfaces and droplet size.