The adsorption of a bond fluctuation self-avoiding walk polymer on an attractive homogeneous flat surface at temperature below the critical adsorption point is studied using dynamic Monte Carlo simulation. Results show that the apparent size R_g,xy² of the polymer parallel to the surface increases exponentially with time during the adsorption process. The relaxation time for R_g,xy² reaching its asymptotic value σ_eq decreases with the increase in the polymer-surface attraction strength E_ps, whereas σ_eq increases with E_ps, indicating that the polymer is adsorbed faster and becomes more extended at stronger adsorption. The polymer's asphericity A_xy parallel to the surface is sensitive to intra-polymer interaction and its behavior is different from that of R_g,xy². Simulation results also show that the two-dimensional behaviors of R_g,xy² and A_xy are different from that of the three-dimensional conformational size R_g² and asphericity A during the adsorption process. During the adsorption, the surface contacted monomer number M increases with time, but R_g² and A show novel behavior as they first increase with M at small M, then decrease with M at moderate M and finally increase with M again at large M. Whereas R_g,xy² and A_xy first decrease with M and then increase with M during the adsorption.