How to transpose
How and why to transpose music
What does transposing mean and why do people do it?
To transpose means to take a piece of music and put it in another key. There are several reasons one might want to transpose, and one of the most common is because the piece of music/song is too low/high for a vocalists voice range. For example, if a song is in C Major, and the singer finds that the vocal part is slightly higher than comfortable, he can transpose the piece a few semitones down to suit him, for example he could take the piece to Bb Major, two semitones down. Also, a Jazz musician might want to perform a Jazz standard in a different key than the one written, maybe because he finds another key more comfortable or to make it easier for a horn player that plays with him etc. But the most important reason, in my opinion, for any Jazz musician to be able to transpose is to be able to practice licks/scales/solos/pieces/everything in all 12 keys.
How to transpose
Let's take as an example the first few bars of the Jazz standard, Autumn Leaves:

There are different ways of thinking when transposing music. One of them is looking at the key signature. In this case, the key signature is two flats, which means the piece is in the key of Gm (relative minor of Bb Major)*. If you notice, there are numbers on top of each note. These numbers represent the scale degree of each note, i.e. the Eb is the 6th degree in the Gm scale (G A Bb C D Eb F G), the G is the 1st, the F# is the #7th etc. So, when thinking in a key center manner like this, all you have to do is decide which key you want your piece transposed in and then follow the numbers to write the equivalent notes. As an example, I have transposed a couple of bars from the above melody into the key of Bm:

You will notice that the key signature has changed to two sharps (the key signature of Bm/D Major) and that both the notes and the chords have been changed. The chords are transposed in the same manner; the first chord in the first example was Cm7, which was the 4th chord in the Gm scale. So when transposing to B minor, it is replaced with the 4th chord in the Bm scale.
Another way
Another way of thinking when transposing is to think in terms of intervals and/or semitones. So, instead of counting scale degrees, you count how many semitones apart the two keys are; the original key and the one you want your music to be in. In this case, the original is in Gm, and lets say we want to transpose it to Am. G is two semitones (or a whole-tone) below A (G - G# - A), so just take all the notes/chords and count two semitones up, to the new notes/chords:

First note was G, counting two semitones up... G - G# - A, so the new note is A,
Second note was A, counting two semitones up... A - A# - B, so the new note is B, etc.
Remember: Transposing can be confusing at the beginning, but it is definitely worth mastering; after all, one of the best practices you can do to improve your playing is to practice stuff in all 12 keys. I hope this lesson helped.
* There are several ways of distinguishing whether a piece is written in the relative minor or major scale, even though the two have the same key signature. One of them is looking at the melody or at the ending chord where the melody resolves.
