What is transposition? Well, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval. This means you could play or sing the same piece in another key by transposing it. Most people do this all the time, perhaps without knowing, when they sing a favorite song on two different occasions. All the intervals between the notes in the song will be the same, so we recognize it as the same song, but if it starts on a different pitch, like A instead of C, it's been transposed.
Transposition like this happens a lot in music for classical singers, since we have so many kinds of voices and ranges of notes each voice can sing beautifully. So the same song might have sheet music published in three or more different keys for various voice types. Here's an example of a melody first shown in D major on the top and then shown transposed to B flat major on the bottom.
Fun fact, this example is from a music theory book written in 1865. It also counts as transposing if you move a melody into the same key, but into a different octave. This happens all the time in vocal music, when male and female singers want to sing the same stuff. Most men can't sing in the same octaves as female opera singers, though a few can, so they'll have to sing things an octave, or even two, lower.
Another example can be found in César Franck's famous Violin and Piano Sonata, which is available in arrangements for several other instruments and piano as well. Here's part of the original violin part. Now here's a transcription of the same bit, except for the last measure, for a cello to play along with the same piano part.
Notice that the key signature has not changed, but the clef certainly has. How many octaves apart are these two versions of the melody? Yes, the cello part is two octaves below the violin part. This is the most straightforward type of transposition, even if the difference is more than two octaves.
The note names are still the same, we just see and hear that one is much higher or lower than the other.