Friction stir spot welding (FSSW) was performed on 2024-T3 Al alloy to study the effect of pin and shoulder geometry (cylindrical and triangular shapes) on stir zone (SZ) characteristics and mechanical properties of welded samples. The process was conducted at a constant dwell time of 5 s and shoulder plunge depth of 0.3 mm. The rotational speed varied in the range of 630–1600 rpm. Based on the observations, the change in the pin shape led to changes in the SZ area, hook geometry, and mechanical properties of the joints. At low rotational speeds, the strengths of the joints made by the triangular pin were higher than those formed by the cylindrical pin. On the other hand, the change in the shoulder geometry from cylindrical to triangular reduced the joint strengths. However, the effect of pin geometry on the strengths was more than the shoulder geometry. Moreover, the strengths were directly related to the SZ area. Scanning electron microscopy of the welded sections revealed formation of cracks in the SZ as a result of S phase local melting at the tool rotational speed of 1600 rpm.