A transposing instrument is a musical instrument whose music is written at a pitch different from that pitch that it sounds when played, the written pitch being a transposition of the actual sounding pitch.
One of the great inventions of western music is a notation that describes the sound that is required (pitch, duration, volume) rather than what you have to do with the instrument to produce the sound (for example, "stop the E string with your second finger on the third fret"). This means that the same piece of music can be picked up and played fairly easily on a whole range of instruments: violin, flute, oboe, piano, guitar, voice, glockenspiel. This facilitates the ensemble playing that is such a feature of western music, and means that you can look at music written for any instrument and know roughly what it will sound like when played, without having to know the details of how to play the instrument.
There are, however, some problems with this system.
It is more difficult to learn to read this sort of music because the player has to learn a mapping from the pitch notated to the fingering (or whatever) of the instrument.
Some instruments have a range which does not lie within the normal clefs. In this case the music is written either an octave lower than it sounds (for example piccolo, descant recorder) or an octave higher (e.g. double bass, contrabassoon, bass guitar). These are then technically transposing instruments, but people tend not to think of then as such, because they don't need much special consideration. A double bass can play cello music, and the fact that it sounds an octave lower just feels like part of the different 'sound' of the double bass.
A more serious problem arises when there is a family of instruments of different pitches. For example the normal (soprano?) flute is pitched in C and is non-tranposing, i.e. when the flute player plays a written C the note you hear is a C. The alto flute is basically the same instrument, but longer, and hence pitched lower. The fingering that would on a normal flute produce a C produces a G a fourth lower. If alto flute music were written at actual pitch, the poor flute player would have to re-learn the mapping from written pitch to required fingering. The alto flute is not in common enough use to make this worthwhile.
What happens instead is that the music for the alto flute is tranposed up a fourth so that if the player reads it and fingers the alto flute exactly as if they were playing the written note on a normal flute then the right pitch of note is heard. The net effect is that the flute player can swap happily between instruments, but the composer or arranger has to specially transpose the alto flute parts. The alto flute is thus a 'transposing instrument in G', sounding a fourth lower than written.
A similar thing happens in other instrument families. For example, clarinets come in various sizes and hence pitches (A, Bb, C, Eb), but the music is transposed appropriately for each size of instrument so that the player can happily move from one to the other. The only complication is that the most common sizes of clarinet are Bb and A; this means that people tend to say 'the clarinet is a transposing instrument' which, while broadly true, is not the case for the C clarinet.
Other common families of tranposing instruments are: