The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on.
Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab.
Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.
Pythagorean comma is the difference between enharmonic notes in Pythagorean tuning, e.g., difference between Ab and G#. Not what I was talking about.
With any 12-tone system, you might have eleven fifths which are 700-x cents wide, on average... and then a diminished sixth (rather than a fifth), which is necessarily 700+11x cents wide as a result. The "wolf interval" is the diminished sixth (commonly G# to Eb, or C# to Ab). In equal temperament, the diminished sixth is equal to a fifth. You might also call something other than a fifth a wolf interval. Basically, an interval which normally sounds good, except you chose an enharmonic version of it, which sounds bad.
"The" wolf interval is the quarter-comma meantone diminished sixth, which is 2^7 / 5^(11/4) = 1.5312. The Pythagorean wolf interval is 2^7 / (3/2)^11 = 1.4798. It's equal to a perfect fifth, minus the Pythagorean comma.
https://en.wikipedia.org/wiki/Meantone_organs_in_North_Ameri...
It was Werckmeister who popularized and advocated systems which don't have wolf intervals (so-called "well temperament", as in "Bach's well-tempered clavier". The composers who cared about wolf intervals generally predate Bach, to give you an idea.
Nowadays (since the 1980s) I can press a button and put my synthesizer or electronic piano in whatever tuning I want, so I can play with quarter-comma meantone temperament without finding one of the organs on the list, and experience it myself.