700c (ISO 622) is the wheel size (other common wheel sizes are 26 inch (ISO 559), 29 inch (which is really the same as 700c), 650b, etc). The second number determines the tire size (via width-ishness). 

See [this][1] page on tire sizing - there is some variance even among tires marked the same thing (and strangely enough, numbers mean different things when written as decimals or fractions), even if the tires are of the same style (say, slicks; Obviously if one tire is knobby and the other is slick, you can't reasonably expect them to have the same tire clearance when mounted in the same bike). Generally, the bigger the second number, the bigger the tire  (knobs can cause this to be violated) and typically more cushioning. However, if its too big, it won't clear the brakes or the fork/frame. 

Typically, your tire should between 1.5x and 2x the inner rim width - too narrow and the rim can take some hits. Too fat and the brakes drag on the tire/handling can be compromised. 

If you haven't bought a tire before, take the bike to your LBS and try on some tires so you're sure they'll clear the frame and are not too wide for your rims. 


  [1]: http://sheldonbrown.com/tire-sizing.html