An octave is the distance between two musical notes that have the same letter name.
If a musician sings or plays a scale (‘do-re-mi-fa-sol-la-ti-do’) the first and last ‘doh’ are an octave apart. It is called an ‘octave’ because there are eight notes in a scale (‘octo’ is Latin for ‘eight’).
If a musician sings or plays the first two notes of the song “Somewhere over the rainbow” (i.e. the word ‘Somewhere’) these two notes are an octave apart.
Two notes that are an octave apart sound very similar, almost like the same note. The scientific reason for this is that the top note vibrates twice as fast. For example: if someone plays Middle C on a piano, a hammer hits the three Middle C strings inside and makes them vibrate 256 times a second. The C an octave higher will vibrate 512 times a second.
Young children who learn the piano will not be able to stretch an octave with one hand, if their hands are too small. Most adults can stretch an octave easily (playing one note with the thumb, and the other with the little finger). Advanced pianists can practice scales in octaves. Violinists can play scales in octaves, too, playing on two strings at once, but this is very hard.
See also: Interval (music)