Dictionaries (at least in the US) seem to consider "gage" to be an alternate spelling of "gauge". But common usage suggests "gage" refers to sheet metal thickness, gage blocks, and most concrete measurement systems and devices. "Gauge" seems to refer to pressure gauges and level gauges. Comparing Machinery's Ready Reference, Machinery's Handbook, and the MSC catalog, there is no agreement on "wire gage" versus "wire gauge". Time seems to have corrupted any clear distinction, and things are most likely different in England and other places.
So my particular preference is to only use "gauge" for devices that measure more abstract things like pressure and level.
Gage: An arbitrary assignment of numbers to size, used on sheet, wire, and many other things (tubing, shotgun bore, needles, and so on). Unfortunately there are so many different standards that using gage to specify a material is almost useless, and perhaps even dangerous. If in doubt, use decimals.
For example, in U.S. gage, the standard for sheet metal is based on the weight of the metal, not on the thickness. 16-gage is listed as approximately .0625 inch thick and 40 ounces per square foot (the original standard was based on wrought iron at .2778 pounds per cubic inch; steel has almost entirely superseded wrought iron for sheet use, at .2833 pounds per cubic inch). Smaller numbers refer to greater thickness. There is no formula for converting gage to thickness or weight.
Wire might be specified in American or Browne & Sharpe, Birmingham or Stubs, Washburn & Moen, Trenton Iron Co., Stubs' Steel, Imperial Wire gage, music wire gauge, or others. In the US, I've commonly seen electrical wire in AWG (American Wire Gauge); even-numbered values are much more common than odd ones. As an example, 12-gauge wire is .081" (AWG), .109" (Birmingham or Stubs), .105" (Washburn & Moen), .028" (music wire gauge). In most cases a larger gauge means smaller wire, except music wire goes the other way around!
If you need specific values for specific gages, many are listed in various references such as Machinery's Handbook.
On September 20, 1994, Rowland Carson posted a nice list of most of the gauge systems in use worldwide:
Alhoff & Muller Music Wire Gauge music wire American Screw & Wire Co Music Wire Gauge music wire, spring wire American Zinc Gauge zinc sheet ASA B32.1 uncoated thin flat metals USA Birmingham Gauge BG iron hoop & strip, steel sheet Birmingham Wire Gauge BWG iron & steel telephone wire Birmingham Wire Gauge for Silver & Gold silver & gold Brown & Sharpe Wire Gauge B & S non-ferrous sheet & strip Brunton Music Wire Gauge music wire Card Wire Gauge CWG fibre yarns Continental Zinc Gauge sheet & strip English Music Wire Gauge music wire English Zinc Wire Gauge zinc sheet Felten & Guilleame Music Wire Gauge music wire German Sheet Gauge sheet metal Instrument Wire Gauge instrument wire ISO Metric Preferred Series (R388) all applications Junge de Paris copper wires Lancashire Pinion Wire Gauge LPG cast steel drill rods Mathieson & Hegeler Gauge M & H zinc sheet Millimetre Wire Gauge steel wire National Wire Gauge NWG steel wire Poehlimann Music Wire Gauge music wire Printers Wire Gauge printers wire Roebling Wire Gauge steel wire Sewing & Darning Needle Series sewing & darning needles Standard Wire Gauge SWG wire & strip, tube thickness Stitching Wire Gauge Stubs Iron WIre Gauge iron wire Stubs Steel Wire Gauge steel drill rod Tinplate finished tinplate Twist Drill & Steel Wire Gauge drill rod US Galvanised Sheet Gauge GSG galvanised sheet US Manufacturers' Standard Gauge MSG iron & steel sheets & plates US Standard Gauge USG stainless steel US Steel Wire Gauge USSWG steel wire Washburn & Moen Gauge WG steel wire Warrington Wire Gauge steel wire Yorkshire Wire Gauge steel wire