Fuji 4130 bike with ARAYA Japan rim 26x1.50 HE VP-20 Old tires are dryrotted. ( old tire code reads 26x1.80 c-863) have no idea what to buy to fit. (rim measures 22" diameter, roughly)
26 means 26 inch and is the nominal diameter of rim and tire. It basically means the smaller type of wheel used on older MTBs, with a rim diameter of 559mm. ('27.5' - 584mm and '29' - 622mm being the other sizes in more common usage now.) 1.80 is the nominal tire width in inches also.
Some MTB tire manufacturers still use the inch measurements, some use the ETRTO size system which uses millimeters.
You definitely need a 26'/559mm MTB tire, but you have choice in tire width as a range of widths will fit on a given rim width. The safe choice is to stay with 1.8", but you could go a little wider if you frame and rim brakes have enough clearance.