I can answer this for the brass more convincingly than for the woodwinds, perhaps someone else can fill in those blanks.

Back before the valve was invented, composers would write music in a certain key. The trombone has a slide which enables it to change length at will, meaning it can play in Eb as easily as in F. But the Trumpet and Horn couldn't change their length as easily, meaning they only had the notes in their overtone series to work with--major chords built upon the fundamental, mostly. In order to play more than a note or two in any other key, the music had to either be written REALLY high--where the notes of the overtone series get closer together--or a horn of a different length was needed. So horn and trumpet players would need several horns to play in any piece that changed to keys that were not closely related.

Now in order to make things the music easier to read, the tradition was that the fundamental pitch of each horn (F on the F horn, Bb on the Bb horn, etc) would be notated in music as a C. This made it so that the player could easily just pick up a longer or shorter horn, and the written music would look the same, and the physical sensation of playing the horn--such as distance between overtone notes--would be the same (or nearly so), but notes of the correct key would be coming out of the bell.

As instrument design progressed, horns and trumpets came with extra tuning slides of various lengths, so that a whole new horn was not needed to play in other keys, but rather you could take a short tuning slide out and put a longer one in, putting the horn in a different key. These slides were called crooks because by the time of Weber and Wagner, composers were changing keys so quickly and fluently, that the slides had to be shaped with a crook in them to hang over the arm of the performer to facilitate quicker slide changes.

With the invention of the valve in 1815, and its subsequent adoption/popularization, this practice fell out of favor (much to Brahms' chagrin: he loved the sound of the natural horn, and hated the valve horn). So the tuba doesn't have this tradition of transposition, but rather it reads the notes on the page as they are, and the player will play that note based upon the pitch of their instrument. The trumpet and horn, though kept the tradition of transposition even after the invention of the valve. The Bb trumpet became standard. Modern players play a lot on both Bb and C trumpets and need to be able to read many different transpositions on both instruments. The horns have kinda settled on the F horn, but need to be able to read in any transposition.

However, one caveat must be included here. When the valve came to be, inventors like Adolph Sax, created whole families of instruments that were meant to sound the same over the entire range of music. The saxophone family alternates Bb and Eb instruments (Bb for soprano, Eb for Alto, Bb for Tenor, Eb for Baritone, Bb for Bass, etc). This made it so that a player of one instrument could pick up another instrument and play the same fingerings for the same written music and be okay. The third space C is fingered the same on all the saxophones, and depending on which saxophone it is, you'll get either a Bb or an Eb coming out of the instrument.

Sax also invented the saxhorn, which accounts for euphoniums, alto horns, and some tubas, also alternating (Eb for alto/tenor horns, Bb for baritone/euphonium, Eb for bass tuba, BBb for contrabass tuba, etc). This is particularly useful in the British Brass Band world, where everyone (except the bass trombone for some reason) reads treble clef parts in Bb or in Eb. Again, the idea here is that a euphonium player can move to tenor horn and read the music the same way with the same fingerings and be in the right key without having to think about the key of the piece AND the key of the horn AND the fingerings that go with both.