Once you have an AWG table with diameters handy, you need to make the actual measurement.
For stranded cables and larger wires, vernier or digital callipers are suitable. If the wire is insulated, cut round the insulation and pull it partially off (or cut round a second time and peel off the insulation between the cuts) to expose a ring of the bare wire to make the diameter measurement on, with the end still trapped in the insulation so the lay of the strands isn't disturbed. Avoid twisting the exposed section as you work.
For fine wires you need a micrometer. For fine stranded wires, count the strands and measure a few individually and take the average. For greatest accuracy, magnet wire should be chemically stripped to remove the enamel.
You can get AWG gauge plates, which are useful for larger wires and cables when great accuracy isn't needed, but make sure you get one from a reputable supplier that is actually marked as being for AWG wire, as there are many cheap ones on EBAY and similar which are either misdescribed as AWG gauges or have incorrectly sized gauging notches.