A "58 (or 580) round" or "60 (or 600) round" grip size refers to the inside diameter of the grip uninstalled.
Both 58 and 60 (referring to 0.580" and 0.600") core sizes are considered standard sizes and most people consider the difference to be rather insignificant. Note that core size is different from outer diameter size. The core size is the INSIDE diameter of the grip and usually has no relation to the OUTSIDE diameter of the grip (which is typically denoted with the sizing terms "Standard", "Midsize", "Oversize", etc.). Some grips only come in the 60 standard core size. If you install the 58 grip on a 0.600" butt diameter shaft, it will be very slightly bigger (1/32") than installing a 60 grip, but most people cannot tell the difference.
The majority of golf grips come in only 60 core diameter size. This size most closely matches the majority of all outer butt diameter shaft sizes. Some grips however come in additional core sizes, like 58 (0.58"), 56 (0.56") or even 62 (0.62").
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