The octave is the most fundamental interval in music. The human ear tends to hear both notes as being essentially "the same". For this reason, notes an octave apart are given the same note name in the Western system of music notation - the name of a note an octave above A is also A. This is called octave equivalency and in some ways is similar to enharmonic equivalency and to a lesser extent transpositional equivalency and, less still, inversional equivalency, the latter two of which are generally used only in musical set theory.
As well as being used to describe the relationship between two notes, the word is also used when speaking of a range of notes that fall between a pair an octave apart. In the diatonic scale, this is 8 notes if one counts both ends, hence the name "octave", from Italian for 8. In the chromatic scale, this is 13 notes counting both ends, although traditionally, one speaks of 12 notes of the chromatic scale, not counting both ends. Other scales may have a different number of notes covering the range of an octave, but the word "octave" is still used.
The notation 8va is sometimes seen in sheet music, meaning "play this an octave higher than written." 8va stands for ottava, the Italian word for octave. Sometimes 8va will also be used to indicate a passage is to be played an octave lower, although the similar notation 8vb (ottava bassa) is more common.
See also: Solfege