0/0 is not equal to 1, it is indeterminate, meaning its value is not known. You may have seen some "proofs" that seem to say that 0/0 is equal to 1, or some other number, but these proofs have a mistake- likely, they divide by zero. Division of any number (0, 1, 75, Infinity) by zero never produces a defined result.

However, if you are considering limits, you may encounter something that at first glance looks like 0/0, but in fact has a defined value. An example:
What is the limit of x/x, as x approaches zero?
If you plug in x = 0 to both the numerator and denominator, you find 0/0, which is indeterminate. However, if you consider that the numerator and denominator are always equal and change at the same rate as they approach (but are not quite equal to) zero, you find that the limit is 1. Formally, this technique for dealing with limits that appear to be indeterminate is known as L'Hopital's Rule.