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Arbor diameter is the diameter of the shaft that the blade mounts on. It's important in that you want a standard size arbor that blades are commonly available for. If your arbor is SMALLER than the hole in the blade, you CAN get an adapter bushing (some blades come with the bushing). Bushings are an unnecessary pain and I would avoid it by buying a tool with the common-size arbor.
10 inch miter saws/blades are mostly generally 5/8" arbor, as BadgerDave said. 12 inch saws mostly have 1" arbor, but some have a removable sleeve on the arbor so that you can mount 5/8" center hole blades as well.