Because it is Christmas and so much of our secular Christmas music is jazz-influenced, here are some thoughts about jazz chords that might be helpful.
First, a caveat. Due to the democratic nature of jazz, there are plenty of musicians that may disagree with me, and most of them will be more qualified than me to answer the question. But I will answer this question with what I have learned from some incredible musicians over the years.
Here is the short answer: there is not really any such thing as a jazz chord. Jazz uses the same chords everyone else uses. That includes the common I, IV and V chords, the minor ii, iii, and vi chords, and the secondary dominants. There are a few “special” chords outside of those that you will see in jazz but you will see them in other music as well. There are flavors of jazz (such as experimental jazz) that stray from these chords just as modern classical music does, but I am not really including that kind of jazz in this discussion.
That being said, jazz gets its distinctive harmonic sound not from the chords themselves but what it does with them. There are two elements that are critical: color and voicing.
By color, I am referring to adding additional notes to the base chords (this is also referred to as “extending” the chords). For example, jazz music commonly uses major 7ths on I and IV chords while traditional and pop music tends to see those chords as simple triads. The Christmas Peanuts music that we all know and love is a great example of using major 7ths.
Here is an example from Seasonal Spice where I use the major 7th in a similar way. Note the FMaj7 in bar 1 with the E natural as the melody note:
The minor chords are also extended. You rarely would play a minor triad. Typically, you add the minor 7th to it and additional notes as well. Note in the example above how I add a minor 7th to the Emin chord in bar 1 and the 7th, 9th and 11th to the Dmin chord in bar 2.
Probably the most distinctive chord in jazz is the dominant. It is actually called the color chord because so many extensions and color are possible. Remember, there are actually 6 dominants that are very commonly used: the V7 and five secondary dominants (V/ii, V/iii, V/IV, V/V, and V/vi).
In the example above, note the use of the b9 in the V chord in bar 2. That is just one of numerous options that can be used to extend the chord. Here are some examples:
Remember: jazz musicians would see all of these chords as A7. They just see different colors applied to each. Also note that these extended chords do not include the 5th. The 5th is unnecessary in a dominant chord; in fact, it sounds a bit clunky when included. By the way, you do not see many chords labeled as augmented either. If you see a sharp 5, that is normally considered a flat 13th.
That leads us to the second distinctive that I mentioned: voicing (where the notes are positioned on the keyboard). If you start using extended chords, you learn very quickly that voicing is critical to making them work. An A7 with a b9 and b13 can be played in numerous positions (voicings) on the keyboard but they do not all work. In fact, some of them will sound horrible.
For this reason, voicing and color are actually almost the same thing. You cannot use color if you don’t understand voicing. And that is why some musicians will interchange the terms.
I am not trying to oversimplify because it actually can get pretty complicated. But these two concepts are really what makes a chord sound like jazz.