Determining Tire Size
Your tire size is composed of 3 numbers. Let's use 205/65/15 tires as an example. 205 is how wide the tire is (in millimeters). 65 is the percentage of the width that will tell you how tall the sidewall is. And 15 is your rim diameter.
Now to figure out how tall your sidewall is going to be you'll need to multiply 205 by .65 (or 65%). You'll find this equals 133.25. So your sidewall is 133.25mm, tire width is 205mm, and rim diameter is 15". Most people don't understand mm or know how many are in an inch. So let's convert that to inches for simplicity (for most people). 1 inch is 25mm so 205 and 133.25 need to be divided by 25 to find the tire width/height in inches.
205/25 = 8.2" wide
133.25/25 = 5.33" tall sidewall
So the tire is 8.2"/5.33"/15". Now when you're changing a tire, the number you're going to be most concerned with are the first and last number. They will be determining whether or not the tire is going to fit on your rim. You can't put an 18" tire on a 16" rim and you can't put a 315mm wide tire on a 7" wide rim. However with width you can put a 285mm (11.4") wide tire on a 10" wire rim. You really shouldn't go more than 1.5" wider than the wheel or 1" narrower than the wheel.
Now don't start thinking you can run any size sidewall and get away with it, you have to use some common sense in that. Running a "the tire is barely there" sort of tire will quickly lead to damaged rims since there isn't enough tire to absorb the shock of hitting bumps and such. And running tires with too big of sidewalls can lead to rubbing issues.