The following table shows the notes required to build some basic major and minor chords:
So, by looking into this table you can understand what it means when a chord is written like this: Am6.
It means:
Am6 = Am (A minor) + major 6th = (A C E) + F# = A C E F#
Strictly speaking, this chord is based on the 1.m3.5.M6 formula, so it should be called: A, minor 3rd, major 6th.
Another example:
Emaj7 = E (E major) + major 7th = (E G# B) + D# = E G# B D#
This chord is also written as: EM7, EΔ7.
Yet, another example:
D9 = D7 + major 9th = D (D major) + minor 7th + major 9th = D F# A C E
Here's an additional table for augmented, diminished and suspended chords:
As you can see, there are several types of diminished, augmented, and suspended chords:
the diminished triad is "dim", also written "o";
the diminished + m7, is also called "half diminished", also written "ø";
the dim7, or "o7", brings the m7 half a step down, and it becomes a 6th.
Most of the time, when you see a diminished written, it refers to the "dim7" chord.
The tables above are, by no means, complete. There exist more chords beyond the ones shown here, but the intention with this page is to provide the general idea of how chords are constructed.