Instruments such as a trumpet and clarinet (tuned to Bb) and alto saxophone (tuned to Eb) require transposing. For example, when the written note C is played on a trumpet, it is the same note as an Bb on a piano. Other instruments requiring transposing include English horn, alto flute, baritone sax, and French horn.
For the pianist, it means they must be aware of the keys of these different instruments. For example, if the pianist is accompanying a trumpet player, they need to realize that communicating with them will be difficult unless they “transpose” notes when they speak of them. If they say to the trumpet player, “Let’s start at measure 14, where you come in on the note C,” the pianist is, in fact, looking at the note Bb on their piano music. This ability takes time to learn. But in the case of a piano and a trumpet, the pianist knows that every note they see on their music is a whole note above what the trumpet is playing. The same care has to be taken when purchasing music. Most sheet music of this type will say something to the effect of, “written for piano and Bb trumpet.”
Seasoned musicians know how to do this and after a few months of training (and performing with a number of different instruments), transposing becomes second nature.
When it comes to a vocalist, there is of course, no transposing needed. For example, a vocalist in the jazz group above, will speak (in terms of transposing and instruments of different keys) the same way as the pianist. The vocalist sings the notes C, D or G in the same way, and on the same pitch, as they are played on the piano. Instruments such as guitar, mandolin, banjo, and recorder do not require transposing.
If the pianist is a composer, these differences pose quite a challenge when composing for a number of different instruments. If the composing is done on paper, large “score” paper must be used to easily see the different parts for each instrument. And every time the pianist writes a part for a trumpet, sax, or other instrument that requires transposing, they have to be adept in writing out the part in a different key, paying attention to all the rules and requirements for that key (such as sharps and flats).
Although learning the different instruments in different keys can be difficult at first, it makes for a very well-rounded pianist and musician.

When is a C not a C? It’s not a riddle or a trick question. It’s
a musical principle. The answer is: a C is not a C when you’re
playing a transposing musical instrument. And it’s not the
instrument that transposes, unfortunately — some arranger or
the player himself must do that.

To understand what this means, you first have to understand the
definition of concert pitch. Pitch, of course, refers to the
frequency of sound. Concert pitch is the universally agreed upon
definition of the pitch of middle C (and subsequently, every
other note in an octave). From the dawn of written music to the
present day, no matter where on earth you are, middle C always
sounds the same. In other words, every musician in the world
understands and interprets middle C in the same way. This is the
simple definition of concert pitch.

Many instruments, therefore, are manufactured to concert pitch.
This means that when multiple musicians read “C” (or any other
note on a scale) on a musical score, they produce identical
sounds. Instruments that are manufactured in concert pitch
include the piano, some brass instruments such as the trombone
and baritone, some woodwinds such as the flute, oboe, and
bassoon.

Many other instruments are not made to produce concert pitch
sounds. These are referred to as transposing instruments. Many
woodwind instruments and some brass fall into this category.
When a transposing instrument player reads and plays “C” in a
musical score, it sounds different from a “C” played by a
non-transposing instrument.

Therefore, music that is written for a band consisting of both
transposing and non-transposing instruments must be adapted for
individual types of instruments. A typical marching band, for
example, may have the same score of music transposed into two,
three or more different keys.

The clarinet is an excellent example of this. It is a
transposing instrument, typically manufactured in B flat. If a B
flat clarinet and a piano played a “C” at the same time, the
sounds produced would not be the same. In order for a B flat
clarinet to play a note that matches a piano playing a “C,” it
would have to play one whole step higher. In other words, it
would have to play a “D.”

What if the two instruments had to play a song together? Music
for the piano would be written at concert pitch. The same music
would have to be transposed one key higher so that the clarinet
would produce the same sound. Though the two songs appear to be
in different keys, the sounds that each instrument would produce
will match in pitch.

Bb instruments include the Bb clarinet, the Bb bass clarinet,
the Bb tenor sax, the Bb trumpet. Music for these instruments
are written in a key which is a whole step higher than concert
pitch.

Eb instruments include the Eb clarinet, the Eb alto clarinet,
the Eb alto sax, the Eb baritone sax, and the Eb alto horn. Music
for this group of instruments needs to be written a minor 3rd
lower than concert pitch.

F instruments include the french horn and the English horn.
Music for these instruments must be written in a key that is a
perfect 4th lower than concert pitch.

It sounds complicated. You may even ask: why not just
manufacture the clarinet (or other transposing instruments) in
concert pitch? Though it seems this would solve some problems,
there are good reasons that certain instruments produce other
pitches. The B flat clarinet is preferred because it produces a
cleaner, more pleasant sound than a C soprano (concert pitch)
clarinet. In fact, the same is true of many transposing
instruments. So while using transposing instruments make more
work for the composer, in the end it’s all about the product.

Serious musicians are known for their keen sense of hearing.
Since only the best quality sound will do for them, composers
will keep writing scores with both types of instruments in mind.

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 take lots of practice 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 is potentially 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.

To learn more about transposing, go to How To Transpose & Modulate
and How To Think In The Key