A useful source is Sheldon Brown's tire size tables.
Tire size are a bit of a black art with many different sizing conventions that look very similar and are sometimes written wrong, so it's quite hard to give a definite answer.
It would be best if you can find the ISO designation which will look something like 40-584; the first number is the width and the second is the rim diameter in millimetre (more precisely, the bead seat diameter or bsd). The rim diameter (second number) for your new tire must be the same, and the width must be similar (within about 5mm). The ISO should be on all tires, but if not, you can measure the size of the rim.
The key point related to your question is that 650B and 650C (the letter should be at the diameter, not the width, as Nathan pointed out) refers to the French system and gives the outer diameter (not the rim diameter), while the letter is an indication of the tire width.
But to fit it on your bike, you need the same rim diameter. The 650B and the 650C have the same outer diameter, but as the tire itself has different width, their rim diameters are different (584mm for 650B, 571mm for 650C, according to Sheldon's table). So a 650C will not fit on the same rim as a 650B.
Therefore it's better to look for the ISO size. If your existing tire is 40-584, you can probably use the narrower 35-584 or a wider 44-584 or something in between, but it has to be xx-584. To be sure, measure the rim width and use the table on Sheldon's website under "width consideration".