Transposition is changing the key of a piece of music, or changing the notes without changing their relationship. This is often done to make the piece of music easier to play or sing. It's a common practice in bands that don't perform their own material; the singer may wish to cover a song with vocals that are far out of his or her range. Transposition can correct that problem by shifting the key into a range that is comfortable for him or her.
Transposition is also used with instruments. Some instruments (called transposing instruments) are not tuned to the same note; for instance, a Bb clarinet is tuned to a B flat and an alto clarinet to an E flat. Transposition of the sheet music for these instruments ensures that they won't sound discordant when playing with the rest of the orchestra or band.
Transposition may be a simple concept, but it's often tricky to achieve. The easiest sort of transposition -and technically it is not transposition at all, since it remains in the same key - is done by octave - simply moving the piece of music up or down eight steps. This sort of transposition may work for a male singer wishing to sing a female's part, but it does little for transposing instruments or other areas of vocal work. In these cases, it's best to use transposition by either scale degree or harmonic interval.
Transposition by scale degree uses the scale degrees of a piece of music to determine the relationship between the notes. Each note in a piece is assigned a scale degree (tonic, dominant, subdominant, mediant, submediant, etc.) and the same scale degrees are used for the new key. This type of transposition once understood is relatively simple, as the relationship between the notes will always remain the same, regardless of the key.
Transposition by harmonic interval uses intervals as a guide for the transposition. By finding the interval between the dominant notes in the two keys, one can deduce the interval between the all the notes. If the difference between the notes is a major third, then transposition of all the notes will be done by a major third. This type of transposition is also potentially simple but calls for an added carefulness when dealing with accidentals that aren't expressed in the key signature.
The very best way to transpose is to learn to think in more than one key. Most beginners start learning in the key of C, so after awhile they can think in that key -- they know where the notes in that key are, and their fingers can get to them easily. Since every key a person can play in is mathematically the same as every other key, by learning to play in a 2nd key one can learn to think in that key, just as they did in the key of C.
Keys are like languages: if you don't know Spanish, you certainly can't think in Spanish, and when you learn to speak it, you will have to rack your brain for the right word for quite awhile before you begin to think in Spanish. Its the same in music -- there are only 12 major keys in which you can play (in contract to languages, where there are hundreds) -- so if you can eventually learn to think in all those 12 major keys, there is no key left that you couldn't transpose in to.
Practically speaking, however, most people don't need to know all 12 keys -- just the keys in which most songs are written: C, F, G, D, A, Bb, and Eb. If you can learn the other six too, that's fabulous, but you can certainly get by with just those 6 keys, or perhaps even less -- and least C, F, and G -- the "big 3" when it comes to keys.