An octave is the interval between one musical pitch and another with half or double its frequency. Pitch is the musical term for frequency.

Music Theory began with the discovery that tones an Octave apart sounded very good together (they were in HARMONY). Those tones (pitches/frequencies) could be written as a ratio of 2:1. Other tones with simple ratios also sounded good together. The ratios of 3:2 and 4:3 were the next best sounding ratios. Using these tones, the Octave was divided into smaller segments of tones. Early music included these 3:2 and 4:3 tones to harmonize with the melody when the music included more than one voice or instrument.

Current Music Theory divides Octaves (in the Western World) into 12 notes. These 12 notes are the CHROMATIC Scale, and can be seen as a piano's white and black keys from one Octave up to the next. Music Theory also divides Octaves into 7 notes (in the Western World), called the DIATONIC Scale. The C Octave begins at C and goes up to the next C, and the note names in it are C D E F G A B. They are the white keys on a piano, and those note names are repeated up and down the piano keyboard. Middle C on a piano is the white key to the left of the middle pair of black keys. The A above middle C has been defined as a note played at the frequency of 440 cycles/second (or Hz, for Heinrich Hertz). Each of the 12 Chromatic notes frequency in an Octave is an equal logarithmic division of the frequency of each Octave (logarithmic because octave frequencies are the adjacent octave frequency multiplied or divided by 2). This is called Equal Temperament.

The modern piano keyboard normally has 88 total keys from a low A to a high C, and slightly more than 7 Octaves of white keys. C is always the key to the left of any pair of black keys.

The black keys on a piano can be named either with the lower note name and a SHARP sign (#), or the upper note name and a FLAT sign (b). These notes are called ACCIDENTALS. A SEMITONE is defined as the Interval between notes in the Chromatic Scale. A TONE is two Semitones.

The bottom note of the Octave in which music is played is named the KEY of that music. The Keys were named after the 12 Chromatic notes. They were important before Equal Temperament because of a subtle difference in sound between different Keys when using earlier Temperaments (see Musical Temperament in Wikipedia). Now, with Equal Temperament, that subtle difference is gone, and that reason for music to be written in different Keys is gone.

Keys still play an important part in Music Theory. Modulating from one key to another makes music more interesting.

If you simply change the Pitch of all the notes in a piece of music, you'd have to add or remove Flat or Sharp signs in front of all those notes that become or were Accidentals. Changing the Key when you change the Pitch of music (TRANSPOSING) allows you to simply show which notes are Accidentals at the beginning of the written music. This notation is called the KEY SIGNATURE.

Beside dividing music according to Octaves and Keys and Chromatic and Diatonic notes, Music Theory (in the Western World) also divides music into MODES. We usually hear music in the Major Mode, and sometimes in the Minor Mode. Each Mode has 7 white piano keys as its Diatonic Scale, and each Mode begins on a different white piano key.

The following are the 7 Modes with the names of the notes they begin on (the white piano keys):
   C ionian - MAJOR Mode
   D dorian
   E phyrgian
   F lydian
   G mixolydian
   A aeolian - MINOR Mode
   B locrian

The Major Mode begins on C and has C D E F G A B as its Diatonic Scale, and the Minor Mode begins on A and has A B C D E F G as its Diatonic Scale. When you play the C Major scale, and then the A Minor scale, they sound different because of how the white keys on a piano are layed out (don't ask why, read music history).

The Key of C in the Major Mode and the Key of A in the Minor Mode don't have any Accidentals in their Diatonic Scales because C and A are defined as beginning their Modes. Other Keys in each Mode will have different Accidental notes because each Mode starts on a different white piano key, and the Semitone spacing of the Diatonic notes of each Mode is different. Keys have to maintain the Semitone spacing of the note that the Mode begins on to maintain that Mode's basic sound. C Major's Semitone spacing is Tone, Tone, SemiTone, Tone, Tone, Tone, Semitone. A Minor's Semitone spacing is Tone, SemiTone, Tone, Tone, SemiTone, Tone, Tone, Tone. Look at a piano keyboard to see this.

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CHORDS provide harmony in music. A Chord is made up of 3 or more notes played together. If you play a C and the E and G above that C, you'll hear a C TRIAD Chord. The reason that every other Diatonic note is used in Chords is simply because they sound best with each other.

Chord PROGRESSION is the choice of the appropriate Chord to follow the preceding Chord. Advanced Music Theory has plenty of good choices for Chord Progressions.

Chords can be built using any note in a Key's Diatonic Scale as the bottom note.
This bottom note of a Chord is called the ROOT. 
If that note is also the first note in the Key's Diatonic Scale, it's called the TONIC note.
The most basic music would begin with a Tonic Chord and end with a Tonic Chord.
Chords built on notes in a Key's Diatonic Scale can be designated by Roman Numerals.
For example, in C Major:
  C Chord is the I Chord (Tonic Chord)
  D Chord is the II Chord
  E Chord is the III Chord
  F Chord is the IV Chord
  G Chord is the V Chord
  A Chord is the VI Chord
  B Chord is the VII Chord

So C III would designate a Chord in C (Major Mode would be presumed) with E as the Root note.
If E is the Root, then the other two notes of an E Triad in C Major would be G and B (every other Diatonic note).

The Tonic Chord in the Major Mode is very harmonically pure, so when music ends on that Chord, it sounds like a good resolution. When it's played following a dissonant-sounding chord, it will sound even more like a good resolution. There are lots of rules-of-thumb in Music Theory about resolving music to the Tonic (ending it well).

The 5th Diatonic note (the Root of a V Chord) is called a PERFECT FIFTH in the Major and Minor Modes, because it has the second best harmonic sound with the Tonic note (after notes that are an Octave apart). The 4th Diatonic note is called a Perfect FOURTH in the Major and Minor Modes, because it has the third best harmonic sound with the Tonic note.

The V Chord is the next most important chord after the Tonic Chord. C Major's V Triad is G B D.
It leads nicely to the C Major Tonic Triad (harmonically speaking).
The V Chord is also called the DOMINANT Chord.

A SEVENTH Chord is a Triad with the 7th note in the Diatonic scale added to the 1st, 3rd, and 5th notes (e.g. C E G B).
C Major's Dominant 7th Chord (the V Chord) is G B D F.

The Dominant 7th Chord leads even better than the Dominant Triad to the Tonic Triad in the Major Mode because a very dissonant pair of notes exists in a Major Dominant 7th (B and F in C Major).  That pair of dissonant notes is named a TRITONE, which consists of notes 6 Semitones apart. Dissonance resolving satisfactorily to consonance is a big part of Music Theory.

A Major THIRD is defined as an Interval of 4 Semitones (e.g. C to E). It sounds good because of E's (roughly) simple 5:4 ratio to C.
A Minor Third is defined as an Interval of 3 Semitones (e.g. A to C).

An AUGMENTED Third is a Major Third that has its upper note moved up one Semitone.
A DIMINISHED Third is a Minor Third that has its upper note moved down one Semitone.

A Major Triad is composed of the Root note, followed by the note a Major Third up, followed by the note a Minor Third up.
(e.g. C E G)
A Minor Triad is composed of the Root note, followed by the note a Minor Third up, followed by the note a Major Third up.
(e.g. A C E)

A 13th Chord could have its 3rd, 5th, 7th, 9th, 11th and/or 13th note Diminished or Augmented, depending on whether that note is a Major or Minor Third up from its next lower note in the chord. Major Thirds can be Augmented, and Minor Thirds can be Diminished. These choices help make music more complex and interesting.

Chords can be inverted, that is, the bottom note in a chord can be moved an Octave up to make the First INVERSION of a Chord, and the new bottom note can be moved up an Octave to make a Second Inversion, etc. Inversions are used for good Chord Progression by making each note in each part of the chords follow nicely from the previous one. In a 3-part harmony, the top, middle, and bottom notes of the Chord Progression all should go well (melodically) with their preceding notes. This is called VOICE LEADING. J.S.Bach was very good at it.

Some notes in chords can be left out without changing the sound much. A note in a chord can also be played an Octave higher or lower. This is called DOUBLING.

The notes of chords can be played together, or they can be played one after the other (up or down). This would be called an ARPEGGIO.

Changing from one Key to another in music is called MODULATION. The best way to Modulate from one Key to another in the Major Mode (only) is to first get to the Dominant 7th of the new Key. From there, it's a natural progression to the Tonic of the new Key. Modulation is more complex in the Minor and other Modes, but once again, Music Theory is full of advice.

If you select Advanced >Chords in PianoRollComposer, you'll see a Circle of Fifths, which is comprised of the 12 Keys. In this circle, any Key is the Dominant note of its counterclockwise neighbor, and each Key has one less or more Sharp or Flat than its neighbor. Neighboring Keys have the most Diatonic notes in common, and thus are easier to Modulate between.

Advanced Music Theory can get complicated, although fundamentally, it's still about the harmonic sounds of notes when played together, which can be worked out by ear.

Counterpoint is an area of Music Theory that's as important as Harmony. It sets out rules-of-thumb for writing music played (or sung) by different instruments (or voices) that don't necessarily require that each instrument play notes that are part of one chord, but rather that the recognizable melodies played by all instruments go well together, and secondarily that the notes go well with the other instruments' notes (as in a chord). Two or more melodies, that can be recognized as independent and playing at the same time (and complementing each other), make music more interesting.

For a more complete explanation of Music Theory, read "The Everything Music Theory Book" by Marc Schonbrun.