To 'transpose' a piece of music is to change its key. That means that all the chords and melody lines and harmony shift, together, to new pitches. The main reason musicians transpose a piece of music is to accommodate the singer's range. Some people say that each key has a different 'color', or vibe, about it, but personally I've never perceived any difference.
But singers are stuck with their vocal range, and if there are notes in the song that they can't reach, either high or low, the tune needs to be transposed to a key that suits the singer. To do that, we refer to a transposition table like the one above, until we can just do it mentally. That takes a while.
How to transpose a piece of music
The blue row at the top shows the 12 keys of music, the green column to the left shows the 7 notes of the scale and the chords that comprise the key. Capital Roman numeral means Major, lowercase Roman numeral means minor. The italic indicates the half/diminished in each key. (Note that these chord flavors are not necessarily the ones required as deviations from the key are often used by the composer of the piece of music.) Each column is a key, and shows the chords that arise naturally from the major scale. These are the chords you are most likely to find in each key.
To use the chart, find out first what key the original tune is in. As an example, let's say it's F#, an awful key for guitar, as it throws up so many barre chords. Look to the left of the difficult key until you find the key you want to play in, let's say C. Looking to the right won't work, because the new key HAS to be lower in pitch than the original, since the capo will be higher up the fretboard.
Count the number of columns away it is from the original and put your capo on that fret. So in this example, C is 6 columns away from F#, put your capo on the 6th fret and play as if in C. Keep referring back to the original key to see which chords are in the tune, and move along the row until you get to the C column. You'll then see which chord is required. Retain the original quality of the chord, so if it was major, make it major; if it was minor7th, keep it as minor7th. If there is a chord in the tune that isn't one of the seven 'related' chords, then find it anywhere in the table and count the same number of columns to the left as the rest of the tune. That will be the chord to use.
Transposing eventually becomes second nature, as you will begin to see tunes as a series of Roman numeral chords. What's more, you'll be able to hear them as roman numerals. The most important thing to remember is that there are only 8 chord letters and they are in alphabetical order: A B C D E F G. Think of them in a repeating circle so that A follows G. Notice how the key of C is made up of all the natural notes - no sharps or flats. The video at the top of the page (which is the same one as Movie 2 on the previous page) will help you to understand how to use the table.