Based on the three-dimensional finite element (FE) model, we perform the eigenvalue buckling analysis for armchair and zigzag single-walled carbon nanotubes (SWCNTs) under axial compression. The effects of nanotube length and diameter on the buckling modes and the critical buckling loads are systematically studied. The lengths of SWCNTs vary from 1.5 to 30 nm and the diameters vary from 0.3 to 4.5 nm. The buckling modes and the critical buckling loads of different SWCNTs are obtained. The simulation results show that the diameter and the length evidently affect the buckling mode and the critical buckling load of SWCNTs. The critical buckling load decreases to a relatively stable value with the length increasing and in general increases with the diameter increasing. With the increase of length-diameter ratio, the buckling modes of SWCNTs change from shell-like buckling characteristics to space trusses-like buckling characteristics, especially for SWCNTs with small diameters. Therefore, the modeling method based on beam theory in nature can well simulate the buckling behaviors.