How to tell if a new pair of brake calipers will fit my wheel, tire, and frame? I have a Takara Kabuto. It has 700x32 tires (I don't really know what that means). 

I was looking at upgrading the brake calipers. I was looking at these, among others.

How do I tell if they will fit my bike? On the page for the SRAM calipers linked above, it says that they fit 28c tires, but in the questions sections, a person said they'll fit on 32c tires. How do I know if that's correct?

I can't find any size information whatsoever for the Tektros.

How do I make sure that any pair of brake calipers I purchase over the internet will be the right width, length, etc.?
Road brake calipers are built with "reach" dimensions.  The reach is the distance from the center mounting bolt to the pads.  Usually this is a range of values (e.g., 47mm-57mm) as you can adjust the pad vertically within the caliper.  It is important you replace the caliper with another caliper of the same reach. If you do not get the appropriate reach you will not be able to position the brake pads to engage the rim braking surface.  

Generally, there are 3 common sizes for road brake calipers.


39-49 mm - "short" reach - Common on race road bikes, lower volume tires (e.g., 700x23c).  The   Tektro R540 (what you linked) has this reach.
47-57 mm - "long" reach - more common on road bikes that slightly larger volume tires (e.g., 700x32c).
57-76 mm  - "Extra long reach" - less common and found on more boutique frames which are designed to fit even larger volume tires or 700c to 650B conversions.


You need to determine which reach your current calipers are. Given the tire size of your current bike (700x32c), if I were to guess, it is likely option (2) 47-57mm.  

If you peruse Sheldon Brown's web page on brake calipers you can see which brands and model numbers will match this reach specification.  I would however suggest measuring your caliper reach to confirm my guestimate before any purchases.