The notes of a diminished chord are all a minor 3rd from each other. So Cdim (a diminished chord built on the note C) is: C D# F#. To create a diminished 7th chord, add the note A (theoretically in this case, the note A is actually a B-double flat: the diminished 7th interval). To create "fully diminished" chords that consist of four pitches, simply build the chord using minor thirds.
A triad consists of three notes: the root (i.e., the note for which the triad is named), the 3rd and the 5th. Most of the time, the root is played as the bottom-most note. So the chord C means that the note C is the lowest-sounding note, with the notes E and G above it. Any chord in which a note other than the root is the lowest-sounding note is said to be inverted. These inverted chords use a "slash" symbol to show the lowest note. So the chord C/E means "a C major triad in which the note E is the lowest sounding note." Such chords are also called "slash chords," particularly in pop music terminology.
Secondary Dominant Chords:
Secondary dominants usually require a fairly lengthy explanation, but here's the condensed version:
A dominant chord is a chord based on the 5th note of a chosen key. For example, the chord G is the dominant chord of the key of C major. It has two important characteristics 1) it is a major chord, and 2) it likes to move up 4 notes to the tonic chord. Generally, if you take any minor chord, change it to major, and follow it with a chord whose root is 4 notes higher, you have created a secondary dominant. You can use these chords to easily change key, or to simply add an interesting colour to your palette of chord choices.
Modal Mixture Chords:
When we write songs, we are usually in either a major key or a minor one. If we choose major, there are certain list of chords that we will usually use, and we get those chords by building triads on each note of the major scale. For example, if we choose C major, the chords we get by building chords on each note of the C major scale are:
C Dm Em F G Am Bdim
If we choose C minor, the chords we get by building chords on each note of the C minor scale are:
Cm Ddim Eb Fm Gm (or G) Ab Bb
If we are in C major, but "borrow" a chord from the minor scale, we call that chord a modal mixture. A typical modal mixture progression, therefore, might be:
C Fm G C