The music that musicians have in front of them when they play is printed music. The sheet or sheets of paper that contain(s) the written notation of what the musician are to play is called printed music.
Sheet music is a subset of printed music. Sheet music usually refers to a "single sheet" of music; that is, one song or piece printed separately. Printed music includes sheet music but also includes music published in collections. People often confuse sheet music with printed music.
Usually people just say “music” if it is obvious that they are talking about printed music:
George was playing from the music, but it was hard to read because some of the notes were badly printed.
The printed music can be used to make it clear that it is printed music; not recorded music:
The music shop sold CDs and printed music.
If, for example, there is a piece of music for violin and piano, then the pianist will play from the piano part, which will have the piano music written, as usual, on two staves. Above those two staves, printed slightly smaller, will be another stave with the violin part. The violinist will play from a separate part which will just have the violin music. This means that the pianist can always see what the violin should be playing, but not the violinist will not be able to see what the pianist is playing. The violinist may need to look at the piano part sometimes to see how the violin and piano part fit together. It would be difficult for the violinist to play from music with both parts because he would need to turn the pages too often. (Unless the violinist had a few measures of rest right before the page turn, the violinist would miss playing some of his/her part while turning the page.)
A score is printed music of a piece written for several instruments. The music (parts) for each instrument are written above one another on separate staves. A conductor can see from the score what each instrument should be playing and how it fits together. Each player only has his or her own part (the notes that that performer plays) in front of him/her. From time to time, if the instrument has long period where he/she does not play (that is, has a block of rests), the publisher may print, usually in smaller type, a portion of the music another instruments (usually an instrument playing the melody) would be playing to help the performer know what to listen before he/she starts playing. These smaller notes are called cues because it cues, or clues, the performer that he/she is going to start playing soon.)
An orchestral score or full score shows exactly what all the instruments of the orchestra play. If the piece uses a large orchestra with many different instruments, the page must be very tall. The conductor uses the score to be able to see which instruments are playing when. The conductor would cue orchestra members (or sections within the orchestra) when they begin to play. Conductors do much more than this (tempo, dynamics, interpretation, etc.), of course, but most conductors use a score when rehearsing the orchestra or when the orchestra is performing.
A miniature score, "study score," or pocket score is like an orchestral score but much smaller. It will not be big enough for a conductor to conduct from because the print will be too small, but it will be good enough for studying, and it will be much cheaper than a large, orchestral score.
In an orchestral score the order in which the instrumental lines are printed will be: woodwind at the top (flute, oboe, clarinet, bassoon and any other special woodwind), brass (trumpet, trombone, tuba, French horn), percussion and strings (violin, viola, cello and double bass). If there is a choir or solo singers their part is written near the bottom, above the cello stave. The bar lines will usually join up the staves of each family. This makes it easier to find the instruments than if they had been drawn all the way down the page.
A vocal score or piano score is the music for a piece for choir and singers (e.g. an opera) in which all the instrumental parts are printed on two staves so that it can be played on a piano. Some notes will, of course, have to be left out to make it possible to play with just two hands.
A short score means a score where an orchestral piece has been written on three or four staves only. It is something between a piano score and a full score. A composer may write a piece in short score when composing it, and write out an orchestral score later. Short scores are not usually published, they are just working copies while the piece is being composed.