First, what is temperament and why do we need it? Simply put, it was Pythagoras who showed that the octave can be divided into twelve notes, deriving these by using the simplest whole number ratio 3:2 that produces a note different from the original (2:1 produces only the same an octave higher). This new note is an octave and a fifth higher than the original pitch 1:1, and is the third harmonic in the natural series. By tuning pure fifths twelve times, we arrive at (3/2)^12 = 129.7463378906, which mathematicians and computer buffs will recognize as being very close to 2^7, or 128 (in other words, seven octaves). By dividing we get 1.013643265, and it is this 'overspill', .013643265 which is called a Pythagorean (or ditonic) 'comma'. In tuning a keyboard instrument this comma must be absorbed by narrowing one or more fifths, and the systems by which this is done are called temperaments.
Not surprisingly, the first of these is called Pythagorean temperament, and is exactly what we have just seen, the purity of the fifths (except the tempered one) being paramount. It has been in use since the earliest times, and in modified forms (splitting the comma between several fifths) until the era of Bach and beyond.
At this point it might help to draw attention to a useful feature of Pythagorean temperament: while most major thirds are wide by a whole comma (painful, but not unbearable in remote tonalities), the thirds which 'straddle' the tempered fifth are pure. For example, if the fifth D-A is narrowed by a comma, the thirds F-A, C-E, A-C# and D-F# will each 'contain' that fifth within their group of four, and so will be (for practical purposes, see below) pure. This has led to confusion between certain types of temperament, which as we shall see actually start out from opposite extremes...
In the 17th century, and no doubt earlier, alchemists and scholars, both church and abstract philosophers still saw the fifth as a sacred interval, and as its name in the scale 'sol' implies, symbolic of the sun. In the deocentric world of the Middle Ages, it would be surprising if fifths and their inversion within the octave, fourths (ratio 4:3, used in the primitive harmonization known as 'organum') had not reigned supreme.
With the Renaissance, things began to change. Suddenly, man had found his place in the world as an individual, and just as suddenly a tuning system with pure (5:4) major thirds -- the interval (or rather degree of the scale, conformably in English 'mi') which, according to the philosophers signifies 'man's place in the world' -- appeared on the scene.
This new anthropocentric system (which only came to be known as meantone for technical reasons by theorists long after its heyday), in its most usual form tempers each fifth by 1/4 comma. (Strictly speaking a syntonic comma, the difference between three pure major thirds and an octave, if the thirds are to be absolutely pure. Since there is no attempt to 'close the circle', this doesn't matter -- and the two are very close anyway.) Since the final amount of tempering amounts to 11/4 = 2.75 commas, a 'wolf' fifth wide by that amount (usually between G# and Eb) is created, meaning that it is impossible to play in certain keys. Some even temper the fifth by larger fractions of a comma, including 1/3 and 2/7. Although both major thirds and fifths are both worse in these systems, they have advantages when keyboards with 19 or 31 notes to the octave are used, allowing the circle to be closed. Meantone systems using the smaller fractions 1/5 and 1/6 also became popular, the latter being the organ builder Silbermann's temperament famously made fun off by Bach (who presumably was wearing earplugs at the time).
These two systems, meantone and Pythagorean, have one thing in common. They are both called 'regular' temperaments, because each of them only uses one size of fifth (except for the fifth containing the 'overflow').
There is one other system that can be called regular, and that is equal temperament. (Because there is no overflow it can also be called circulating, or closed. This allows modulation to all keys.) It should not be thought that equal temperament is a recent invention. As a theoretical ideal it was already known (and perhaps practiced) in remote antiquity. The problem (particularly with an instrument like the harpsichord which is rich in harmonics) is that everything is equally out of tune. The fifths are scarcely noticeably bad, but the major thirds, while not Pythagorean, are bad enough to be unpleasant if listened to closely.
(Owen Jorgensen goes to some lengths to prove that true equal temperament was in fact not achieved accurately (and perhaps, not even sought) on pianos until tuners started using thirds and sixths to check beat frequencies, which began in 1917 -- conveniently timed to coincide with the arrival of atonal music. Throughout the Romantic era, composers continued to appreciate the differences between key-characteristics afforded by slightly unequal systems, and in some places meantone was still in use even in the mid-19th century.)
The three basic systems above constitute a kind of tri-polar magnet between three philosophical ideals: purity of fifths, purity of thirds and purity of intervallic consistency. Next, I shall outline briefly the ways in which the first two, in the pursuit of modulatory freedom can move in the direction of the third, resulting in the temperaments which, following the German term wohltemperierte are now ungraciously described as "well-temperaments" (an expression perhaps redolent of the currently fashionable "wellness", which used to be known as health). However, modified meantone and modified Pythagorean still retain some characteristics of their basic ideal, despite apparent overlaps and similarities in practice.