Clarinet in B-flat, Graves & Company, Winchester (New Hampshire), ca. 1830-40. The Metropolitan Museum of Art, Open Access (public domain).
A clarinettist reads a C and a B-flat comes out of the instrument. It is not a mistake and it is not a quirk: it is the rule for a good half of the winds in the orchestra. The instruments that behave this way are called transposing instruments, and the reason they exist is the reason certain passages sound the way they do.
The story begins before keys and valves. An early brass instrument, without mechanism, can produce only the notes of the harmonic series of a given length of tube. If you need the same music in another key, you do not change the fingering: you change the instrument, or rather you slip in a different piece of tube, the crook. The composer therefore always wrote in the same way, as if the instrument were in C, and at the top of the part indicated which tube to fit: horn in F, trumpet in D, and so on. The player read the same notation and obtained different pitches. It was an economical and ingenious system, because it spared everyone from learning ten fingerings.
With the woodwinds the principle is similar but the cause is another: the families. The clarinet is not one instrument, it is at least three, in B-flat, in A and in E-flat, with different sizes and colours. By keeping the notation the same for all, the clarinettist learns a single fingering and applies it to the whole family, changing only the instrument in hand. The same goes for the saxophones and for the cors anglais.
When valves and keys arrive, the technical reason largely lapses, but the convention remains, because in the meantime thousands of parts exist written that way and generations of players have been trained that way. Even today a horn reads in F and sounds a fifth lower, a trumpet in B-flat reads a C and sounds a B-flat, a bass clarinet reads in treble clef and sounds an octave and a tone lower, and most of the audience knows nothing about it.
The practical consequence falls on three categories of people. The composer, who has to write each part at its written pitch and hear the real one in his head. The conductor, who in front of an orchestral score reads at the same time instruments that state pitches different from the printed ones, and has to transpose at sight while turning pages: it is one of the skills that separate those who read a score from those who merely look at it. And the copyist, who if he makes a transposition error sends twenty people into rehearsal on a note that does not add up.
There is also a sonic effect that concerns the listener and is the reason the matter is not mere bureaucracy. An instrument sounds better in certain keys and worse in others, because notes that fall on comfortable positions resonate more. Eighteenth- and nineteenth-century composers chose the pitch of the instrument according to the character they wanted: the clarinet in A for certain atmospheres, the one in B-flat for others. It is not a conservatory legend, it is a physical difference in the length of the tube and the position of the holes, and anyone who swaps one for the other out of convenience changes the colour of the piece.
The term to remember, if you must remember only one, is concert pitch. When a score is written in concert pitch, every instrument states what is actually heard. Many twentieth-century study editions are made this way, to make reading more honest. The parts the musicians have on their stands, on the other hand, remain transposed, because changing them would mean retraining entire generations to solve a problem they do not have.