**THIS ARTICLE IS UNDER SOME SERIOUS CONSTRUCTION!
A lot of the time, you'll hear people that claim music theory is not important, or maybe even that it stifles creativity. Often cited are case examples such as Paul McCartney's inability to read sheet music. While it is absolutely true that music theory is not required to write great music, for some people, it can help greatly, and introduce new ways to express oneself through music.

Music theory is a set of concepts and a language you can use to communicate ideas about music with other people. The main benefits of learning about music theory are that you will be able to express musical ideas in a concise and accurate manner and be able to think about music in a structured way.

I. The Notes and Keeping Time
In Western music, there are 12 notes. That's it. Well, not exactly.
There's a special thing with frequencies and our ears (see Section VII) where, when measured in Hz (Hertz, a unit that measures cycles per second--in this case, oscillations per second), a note with twice the frequency as another seems to work perfectly with it, to your brain. The interval between these two notes is called an octave.
For example:
Many standards define the note A as exactly 440Hz.
Doubled, this is 880Hz. According to tradition, this note is also A. As is 1760Hz.
The notes are as follows:
A, A♯ (we will talk about sharps and flats soon), B, C, C♯, D, D♯, E, F, F♯, G, G♯

Note Length
A measure is a unit in music which defines the length of an arbitrary phrase, sort of like a coherent musical statement. These measures can be and are subdivided, into equal intervals, usually halves, quarters, and eighths.
A quarter note is a quarter of a common time measure (see below). As an example, a 7/4 measure can hold seven quarter notes, while a 2/4 only two.
An eighth note simply takes up half the time of a quarter note.
A half note is twice the length of a quarter note.
A whole note takes up an entire common time measure.

In sheet music, adding a single dot after a note increases its duration by half; thus, a dotted quarter note means a note with a length or value of a quarter note plus an eighth note.

Note that these names are conventions that were created for the purpose of writing sheet music, and are not as useful or necessary when writing music in a tracker.

Time Signatures
A time signature is the way to express the number of beats in a measure. The most common time signature, and the one most often used in Western popular music, is 4/4, also called common time.

The top number in a time signature tells how many beats are in a measure, and the bottom tells what length of note makes up a single beat. A time signature is NOT a fraction--3/4 are 6/8 denote two different meters (by convention, 3/4 denotes a measure split into 3 quarter-note pulses, and 6/8 denotes a measure split into 2 dotted-quarter-note pulses).

II. Intervals and The Ionian (Major) and Aeolian (Minor) Modes
Now that we have our 12 notes, we have to figure out how they work relative to each other. A way of looking at relative pitch is examining consonance and dissonance, which are respectively defined as when notes, when played simultaneously, either work 'well' together, or 'clash' with each other.
As a rule of thumb, the closer notes are together, the more they clash, until the difference becomes null to our brain. Notes that are one or two semitones apart are generally said to be dissonant. This has to do with frequency intervals and ratios (see Section VI).
The intervals relative to a base (or tonic) are displayed below. Those italicized are relatively important!

0 semitone difference = unison (the same note)
1 semitone difference = minor second
2 semitone difference = major second
3 semitone difference = minor third
4 semitone difference = major third
5 semitone difference = perfect fourth
6 semitone difference = augmented fourth
7 semitone difference = perfect fifth
8 semitone difference = minor sixth
9 semitone difference = major sixth
10 semitone difference = minor seventh
11 semitone difference = major seventh
12 semitone difference = octave

This is rather confusing at first glance. However, there seem to be recognizable patterns. For example, a minor <x> is always closer to the tonic than a major <x>.
The perfect intervals are the most consonant intervals, other than the octave!

III. Chord Theory
Now that we've learned the two most widely-used scales, we can assign chords to them. A chord is three or more notes played either simultaneously, or sequentially. A chord played sequentially is called a broken chord or an appergio. A chord consisting of a scale's note, that note's minor/major third, and its perfect fifth (usually, at least, the fifth can be flatted or sharpened) is known as a triad.
In Roman relative chord notation, all chords are represented by a Roman numeral. The major chords (which use a major third) are represented by capitalized numerals, while minor chords (which use the minor third) and diminished chords (which uses a minor third and flats the fifth to an augmented fourth) are represented by a lowercase, with the diminished chords differentiated with a small circle next to the numeral.
In a major scale, the tonic chord is represented by a Roman numeral I.

The chords are named as follows:

I = tonic
II = supertonic
III = mediant
IV = subdominant
V = dominant
VI = submediant
VII = leading

We will get into how each chord is used shortly.

The major scale has the following chords:

I ii iii IV V vi vii°

The minor scale has the following chords:

i ii° III iv v VI VII

As you can see, both of these (and all other modes of the diatonic scale) have three major chords, three minor chords, and a single diminished chord. The order is even the same! The only difference is which chords take prevalence in the mode. This leads us to one of my favorite revelations in all of music theory.
A relative key is a key that has the same notes and chords as another key, but the parallel key is just shifted to become major or minor! Case example:
C major consists of C, D, E, F, G, A, and B. Therefore, its chords are:

I - C
ii - Dm
iii - Em
IV - F
V - G
vi - Am
vii° - B°

Now, shifted forward so that vi becomes i, we get A minor!

i = Am
ii° = B°
III = C
iv = Dm
v = Em
VI = F
VII = G

It's almost magic to me! And you can do this with any mode, even into modes that are not major or minor! (See Section V)

A parallel key is a key that has the same root note as another key, but has different notes, and therefore, different chords! C major and C minor are parallel keys!

Chord Progressions and Cadences

Seventh and Extended Chords
A seventh chord is a triad that has had a seventh tacked on to it. There are three main types of seventh chords:

A dominant seventh chord adds the minor seventh of the key to a major triad.
A minor seventh chord adds the minor seventh to a minor triad.
A major seventh chord adds the major seventh to a major triad.

There are also lesser-used seventh chords.

Other Chords of Note

Borrowed Chords and Harmonic Dissonance
You don't just have to use the triads provided to you by a key! A borrowed chord is a chord that is not in the key, but is instead borrowed from another, usually closely-related key. A related key is a key that shares many similar chords to another key. For example, in A major, C and G are not chords in that key!

Borrowed chords are represented one of two ways:

With a flatted Roman numeral (such as ♭III, which means it is a semitone lower than the III chord)

Or with a slash, representing which key the chord was taken from! (such as IV/V, which means you have taken the IV chord from the key in which the V chord is the I chord)

Dissonance can be used in a number of ways in harmony, such as creating emotions of suspense, dread, or horror, or simply as a transition to a more consonant chord.

IV. Melody
Take a note. I will take C, and name it Do in solfege notation. Let's also give it the number it corresponds with in its major mode. With this information, we can conclude we are working in C Major.

Do = 1 = C
Re = 2 = D
Mi = 3 = E
Fa = 4 = F
Sol = 5 = G
La = 6 = A
Ti = 7 = B

V. Uncommon Modes
We know by now two scales! You may have been wondering that if we can shift the major scale a certain amount to become the minor scale, can't we shift it different amounts to become different scales? And indeed, we can!
These other modes are listed below, by the chord from the major scale that becomes I.

ii - Dorian
iii - Phrygian
IV - Lydian
V - Mixolydian
vii° - Locrian

These modes are not as commonly used as their major/minor counterparts, but are more common than one may think, and are often confused by novices for major/minor in various pieces.

VI. The Science of Sound

VII. Timbre and Sound Design

VIII. Miscellaneous Tips and Tricks

IX. Helpful External Links and References
http://www.youtube.com/watch?v=i_0DXxNeaQ0
http://hooktheory.net
http://musictheory.net