Why some instruments read different notes
Clarinet, trumpet, saxophones, and horn are transposing instruments: the note on the page is not the pitch that sounds. A trumpet's written C sounds a concert B♭, so a trumpet part is written a whole step higher than concert pitch. For the full story, read concert pitch and transposing instruments.
Why a key sometimes changes its name
Some transpositions land on keys with seven sharps or flats, or on keys that don't exist, like D♯ major. Parts are written in the simpler key that sounds the same, so the transposer shows D♭ major instead of C♯ major and notes the other name. See key signatures for how to read them.