Motion pictures on film have long been shot at 24 fps, so the 24 fps video rate gives what many consider "film-like" motion. The faster 60 fps rate produces a more "real" feel to some viewers. The standard 30 fps rate works well for general use. Experiment with the different rates your HDSLR provides and see which you prefer.
With some early HDSLRs, the frame rates really were 24 fps, 30 fps and 60 fps. But the "real" NTSC frame rates are 23.976, 29.97 and 59.94, and most of today's DSLRs shoot at those rates rather than 24, 30 and 60 fps. (Your DSLR's instruction manual will tell you which it uses; from a shooting standpoint, there's no difference between 23.976 and 24 fps.) Where did those odd numbers come from? In the days of black-and-white TV, the broadcast frame rate was 30 fps, half the 60 Hz rate of U.S. AC power. When color TV came along, there was interference between the color signal and the sound, so the powers that be decided to reduce the rate to 30/1.001 = 29.97 fps to eliminate that problem. The 23.976 rate is 24/1.001, and 59.94 is 60/1.001.
Note that you don't necessarily have to play back videos at the same frame rate at which they were captured. If you shoot at 60 fps and play back at 24, you'll get a slow-motion effect. Conversely, if you shoot at 24 fps and play back at 60, you'll get a speeded-up effect. Some compact digital cameras can shoot lower-resolution video at 300 fps, even 1200 fps, which is good for special effects when high resolution isn't required.