How do I determine my wheel size Would someone explain wheel sizes.

If I am not mistaken, the wheels on my Trek 520 touring bike are 700C. There are no markings, and of course, no documentation of any kind. I assume 700 means 700mm, so 700 / 25.4 = 27.56 inches.

If I measure the rim however, its diameter is approximately 25.1 inches.

If I buy Continental tires, the size is 28 inch, but I have to buy 27 inch Bontrager tubes.

To make matters worse, per this link, tires come in sizes of 26, 27, and 28 inch, and size 700. 

http://www.schwalbetires.com/bike_tires/road_tires/marathon_420

How can I determine my wheel size, and are these various sizes interchangable. Also, is the 'C' suffix important, and what does it mean?

Of course the tires are marked, but I am under the impression (per the dealers comments) that Continental marks their tires differently than everyone else. Also, the tire markings don't help for ordering a new wheel. 

With so many tire sizes available, how can I know if I am using the correct size.

After reading the various answers here, I should explain that my confusion on this matter stemmed from the fact that I used to work as an auto mechanic. In the auto industry, car tires are based on wheel size, 13", 14", 15" etc, and there is no possible way to mount a 15" tire on a 13" rim for example.

So when I saw all these different tire sizes, I assumed they all required a different wheel size, but in fact, there is essentially just one wheel diameter for my type of bike, which they refer to as 700C, which is actually 622mm. 

I also initially assumed that the C referred to the wheel's width, but that isn't true either. The C is apparently just antiquated terminology.

So at least in theory, I can mount a 26", 27" 28" or 29" tire on my 700C (622mm) rim. I just need to verify the width of the tire is suitable my wheel width, which is what Sheldon Brown's article explains.

Also, I measured my Continental tires, which are marked as 28 inch. They actually measure 27 inches (fully inflated), which must be why my dealer calls their sizing non standard.
I too own a Trek 520 and the bike does indeed come with 700C wheels and has for quite a number of years. 

The labeling of a 700C tire is somewhat misleading due to historical reasons. 

Originally, tires were sold in sizes based on the outer diameter of the tire. 700A, 700B, and 700C tires were all 700mm in diameter. However, the 700C tires were the thickest and the 700A tires were the thinnest. This difference was made up by using a smaller diameter rim for 700C tires and a larger diameter rim for 700A tires.

Nowadays, all tires labeled as 700C mount on a 622mm rim. Most tires will have both labels on them. 622 is the size recognized by the International Organization for Standardization, or ISO but 700C has become so common that it is the de facto industry standard label and is how most people will refer to that size. 

Tire widths are (sort of) an additional complication. This page has a chart showing what widths tires are compatible with what width rims. However, I have run all sorts of widths on my 520, have never worried about rim compatibility, and have never had a problem in that respect. (I have mounted one brand of 700x35 that was big enough to rub on my frame, even though lots of others worked fine.)

All of this information and more is available on Sheldon Brown's tire sizing systems page, which is a fascinating read if you want to dork out over tire sizes for a while. 

The following is in response to your edit:

It is not correct that you can mount a 26" or 27" tire on your 622mm rim. 26" and 27" tires are two different entities and each mounts on a different size rim. A 26" tire mounts on a 599mm rim and a 27" tire mounts on a 630mm rim.

Also be wary of 28" tires. While some mount on a 622mm rim, others mount on a 635mm rim.

When in doubt, trust the ISO number and stick within the widths on the chart mentioned above.