Often I'm asked why certain chords relate to each other and how to derive them from said scale, how to string chord progressions together and why two chords often work together. Without going into Pythagorean tuning and Equal temperament ; there are basically a finite number of chords that work 'well' together. These both sound 'pleasing' and work on deep mathematical levels. The types of chords are the ones you'll hear in 95% of written western music.
From Barbershop quartets to Dmitri Shostakovich, and from the Beatles to Brian Eno, there are 'correct chords' that have been used in their compositions. The reason I'm keep lots of stuff in quotation marks is that there's plenty of examples of popular music that steps outside of these simple rules, but that's not a topic for now.
In short there are 7 chords related to each scale. The ones that appear most commonly are the I, IV and V (you'll see the vi a lot too). In C, this would be C, F, G and Am (respectively). If you're unfamiliar with the Roman Numeral system, it's worth getting aquatinted with, as I will often refer to it.
The chords in C are as follows:
Notice if you read each note of the chord vertically you are reading the C major scale starting from a different position.
Each chord is built from two intervals: major 3rds = 4 semitones and minor 3rds - 3 semitones. Any combination of major and minor thirds gives you ANY chord (well, most of the useful ones anyway!).
The Major chords are built root, major third, minor third, and the minor are built root, minor 3rd, major 3rd. The only exception is the B minor 7 flat 5, which is built root, minor third, minor third.
If you have access to a keyboard and lay these chords out, you'll notice a sort of leap-frog pattern, taking this concept, if you were to add a 4th note to each of these, you'll get this:
Here you'll see Major 7 chords are built root, major third, minor third, major third. Minor 7 chords are built root, minor third, major third, minor 7. The exceptions here are G7, root, major third, minor third, minor third and B minor 7 flat 5, which is root, minor third, minor third, major third.
Once you've nailed this concept, you'll find that chord are really quite simple to work out in any key. In-fact sometime's it's easier to work out a chord than a key then extract the chord.
The above 'leap-frogging' process can take you further still, adding 9ths to chords:
There's a lot of discussion points here, firstly I've reverted to calling the chords what I'd write them as, i.e Cmaj9 rather than C major 9.
Secondly why the hell am I calling them 9ths? This is kind of a weird convention everyone uses it. If you imagine two scale laid one after another, (C D E F G A B C D E F G A B C), if you were to count, the 2nd degree would be number 9 by the time you get to the second octave. With a lot fo extensions, they tend to appear in the 2nd octave. This is why you will see things like Dm9, G13, C11. The quick way to discern what it is, is to minus 7 i.e 9-7=2, meaning 9th is the 2nd degree of the scale.
Thirdly the Em7(b5) seems to buck convention, why is it not Em9? This is because in the key of E minor, the 9th (or 2nd if you're still confused from above) degree would be F#, not F. So what you're doing here is in-a-sense, flattening the 9th. The same applies with the Bm7(b5, b9) chord.