Briefly, to subtract a/b from c/d we first write them in a consistent way. For example, a/b = 2a/2b = 3a/3b = ... and c/d = 2c/2d = ... . What we want to do is find a common denominator (bottom line). So a/b = ad/bd and similarly c/d = bc/bd. So a/b - c/d = ad/bd - bc/bd. But now the fractions are consistent: that is, both are expressed as a number of (bd)-th parts. So we can simply subtract the number of parts, to get (ad-bc)/bd. There may be a common factor on both top and bottom, and if so we should remove it by dividing through.
Example: 3/4 - 2/5. 3/4 = (3x5)/(4x5) = 15/20 and 2/5 = (2x4)/(5x4) = 8/20. The difference between 15 twentieths and 8 twentieths is 7 twentieths, ie 7/20. In this case there is no common factor and 7/20 is in lowest terms.