A sharp (#) raises the pitch of a note by one semitone. A flat (b) lowers the pitch of a note by one semitone. In music notation the # and b signs (called accidentals) are placed before the note heads.
As you can see from this example, the same note may have two different names (i.e. F# and Gb have the same position on the fretboard and are the same note). These are referred to as enharmonic notes. Other examples of enharmonic notes are: A#/Bb, C#/Db, D#/Eb, F#/Gb, G#/Ab. (The same principle also applies to the naming of chords e.g. A# Major = Bb Major, C#m7 = Dbm7 etc.)
With the inclusion of sharps and flats, there are 12 different notes within one octave, e.g.:
Note that there are no sharps or flats between B and C, or E and F because those notes are already a semitone apart.