You can approach the subject of temperament either as a mathematical exercise, or as a practical one (i.e. what it sounds like).
Mathematically, you are trying to find the "best" solution to a problem that is insoluble. Empirically, the intervals of a perfect fifth and an octave sounds "exactly in tune" if the frequency ratio of the notes is 3:2 and 2:1. The problem is that you can't make an exact octave by stacking up exact fifths. If you build a cycle of fiths like C, G, D, A, B, ... you eventually get to an interval C - B sharp with a frequency ratio of 531441:262144, which isn't quite the same as the octave C - C with a ratio of 524288:262144 or 2:1.
There have been almost as many different ways to "make B sharp the same note as C" as the number of people who have written anything about the problem.
The current "standard" for western music is equal temperament, where you split the octave into 12 equal parts. That makes a "not quite perfect fifth" with a ratio of 2.9966142 : 2, instead of 3:2. That doesn't look much different on paper, but it is just about audible.
There were several different solutions to the problem in use in the 18th and 19th centuries, all of which allowed playing in every major and minor key. These "well-temperaments" result in semitones of slightly different sizes, and therefore each of the 24 keys has a slightly different sound - and 18th and 19th century composers knew about that, and used it intentionally.
This is a good demonstration - Bach's organ prelude and fugue in D (BWV 532). The start is basically D major scales and chords, ending with an A major chord. At around 30 seconds this is suddenly followed by passage built around an F# major chord, and the tuning of the F# chord is noticeably different from the D and the A - probably a "WTF???" moment for Bach's original listeners, who were not used to this sort of thing.
There are several unequal well-temperaments that are currently used for "historically informed" music performances. https://en.wikipedia.org/wiki/Well_temperament lists the common ones, and has links to the details of most of them.
Unlike a keyboard instrument, the violin is not restricted to playing in any "fixed" temperament, except for the tuning of the open strings. In practice, individual notes are "tempered" according to the musical context, often by much more than the difference between a well-temperament and equal temperament. I've seen claims that professional players sometimes pitch-shift individual notes by almost a quarter-tone.
In contemporary music (since about 1950) most of the interest has been in equal-tempered tunings but with more than 12 divisions per octave. By dividing an octave into more than 12 equal parts, you can get intervals which are "closer to perfectly in tune" than Equal temperament, but still "play in any key". 19, 41 and 53 divisions are cases that work out well and have been explored musically.