I've read this term many times. However, I'm not quite sure of what it exactly is. What is a transposing instrument? What would be an example of one?
A transposing instrument is one for which the standard practice is to write music in a key different from the sounding pitch of that instrument.
For example, a non-transposing instrument is something like a piano (anything with a keyboard, really)--when you read a C on the staff, you play a C and it sounds a concert pitch C. Most pitched percussion instruments fall into this category, as do traditional C transverse flute, oboe, harp, tuba, and most string instruments (like violin, viola, cello).
A transposing instrument is one where the player reads a C, plays a C, and what sounds is the name of the key of the instrument. Most instruments are often referred to with the key in their name, for example Bb trumpet, Bb clarinet, Horn in F, Eb alto saxophone, Bb tenor saxophone.
There is a third case, where instruments sound an octave displaced from the written notes (to avoid the player having to read too many ledger lines). These aren't usually grouped with keyed instruments, but they are technically transposing (for example: crotales, guitar, string bass, piccolo).
Players of transposing instruments will have trouble reading music written for other instruments in different keys, unless they are experienced at transposing written music on the fly (regardless of transposition, this is also true for instruments written in different clefs.) The reason for transposing instruments has to do with the fact that many of these instruments come in different sizes that are all playable by someone who knows the technique and fingerings for one of these instruments. There are also historical reasons owing to the fact that brass instruments (before the invention of valves) could only play in the harmonic series native to their current instrument.
Anyway, when a saxophone player picks up a saxophone, regardless of whether it's keyed in Bb or Eb, the player will use the same fingering for written C on each instrument. The music needs to be transposed into the correct key for this to work, but the result is that the player can play any size saxophone with the same set of fingerings. If the music was not transposed, the player would have to have a different set of fingerings for each saxophone that they played.
The above applies as well to the various sizes of clarinet, and the various keys of trumpet.
There are, of course, exceptions to this rule left and right--french horn is a particular beast, as professional horn players will see music written in a HUGE range of keys, and be expected to transpose at sight to, typically, an F/Bb double horn.