So, there are nominal size designations such as 700C, 700A, 650A, 29rrrr,27", 28", 26", etc. None of the rims or tires with these designations will have any measurement in any common unit system that matches the nominal size designation except by coincidence. It's sort of like cast iron pipe sizing -- there is nothing about any 1/2" pipe or fitting that measures 1/2", except, of course, a 1/2" length of 1/2" pipe is 1/2" long. The only standard in bicycle tire sizing is the diameter of the bead seat which, for a nominal 700C rim/tire, is 622mm. Your rim is marked 622 which identifies it as a 700C rim as opposed to a 27" rim (bsd = 630 mm) or anything else. To figure out where the bsd is measured from. take the 622 mm/700C rim and measure what the 622 mm diameter matches. Or, look here:http://www.sheldonbrown.com/gloss_ba-n.html#beadseat

If you look at the graphic on that page you can see that you can measure the bsd pretty well by measuring the major diameter of the rim and the depth of the bead seat from that diameter and adjusting. Alternatively, just look at the rim for a marking -- almost all will have one.

As already mentioned, the 18 indicates the inside spacing from flange to flange. Rim manufacturers are not consistent across brands as to how they designate width. Some specify the inside width and others the outside. Typically, there is about 5 mm to 6 mm difference between the two -- your 622 18 rim is probably about 23 mm to 24 mm measured from braking surface to braking surface, unless it's designed for a disc brake in which case it doesn't have a braking surface.

By now you should see that the tires for the rim you have are standard 700C tires. Some people say that the narrowest tire you should use needs to be a little wider than the outside rim width -- in your case that would be a 25 mm tire. That would seem a little narrow on that rim to me. People put really fat tires on rims that wide. There is a table giving recommended maximum and minimum tire widths for various rim widths on the sheldon site. See here for that graphic and a bunch of other really important stuff about tire sizing:http://www.sheldonbrown.com/tire-sizing.html

edit: I see that you have 700C 32mm tires. They will fit those rims perfectly.