So dealing so far with only the four classical western elements: Fire, Air, Earth, Water (no Wood, no Heart, no Boron.) I have found that they correspond to distinct personality types, and that there’s a lot of tropes classifying a group of four people into these. 
What do they mean?
Fire is the most famous. We even call someone “a fiery personality”. It wants change, and is unafraid of the destruction or attention that change will bring. Fire is passion, eagerness, oscillation, variance, ambition. Pros: bold, forthright, gets stuff done. Cons: angry, impulsive, leaving a trail of destruction in its wake.
Water is its opposite in some ways. They “go with the flow”. It’s about adaptation, acceptance, yielding and evolving. Sometimes it’s a passive trait of letting the world change you, other times its actively fitting the new scenario. Water is open to fitting whatever context presents itself, like liquid filling a vessel. Pros: agreeable, constantly changing and growing, fits the situation. Cons: deceptive, two-faced, passive, lacking strong convictions, uncreative. 
Earth is unchanging. Have you ever called someone a “rock”? What it presents the world in one time and place, it presents to everyone. It’s a foundation everyone can build on, and doesn’t let the world change or destroy it, but also doesn’t hide from the world. Pros: stable, supportive. Cons: Blunt, unyielding.
Air is freedom. They wish to be “as free as the wind”. Most of all it does not want to be boxed in or chained down. In many situations it will refuse to engage, or come up with a thousand different ideas, all of which are fascinating but none of which are committed. Every choice made closes a dozen other doors, and if that choice can be delayed to the last moment, then the world remains un-reduced. Pros: open-minded, cautious, independent, non-confrontational, unbounded. Cons: evasive, timid, adrift, uncommunicative.
Earth is formed. Like a solid, it retains its shape, volume, and mass.
Water is form-changing. Like a liquid, it retains its volume and mass, but not shape.
Air is formless. Like a gas it retains only mass.
Fire is form-destroying. Like a reaction, it retains nothing.
First off, if you actually feel a flavor affinity for any specific element, there you have it. People who just really LIKE fire, or water, or what not, I find tend to fit the personality archetype pretty well. This is mostly a nod to all the fire people out there, but I know people who just really like the imagery of air and water as well.
One question that epitomizes the difference: How do you react to an uncomfortable question?
Air: Evade the question, or allow the asker’s assumptions to answer it for them while you say nothing of substance.
Earth: Answer it forthrightly.
Water: Lie… or at least, tell them what they want to hear that may be substantively different from what you’d tell someone else.
Fire: Attack them for asking such a bad question.
(this isn’t absolutely the case, but more, which of these do you do more than most people.)
In talking about this with people, I’ve found these identities show up not so much when just introspecting yourself, but in the arguments you have with others. People of two of the same element may easily see differences between them… but people of two different elements often have very similar arguments, across different subjects and participants.
Air tells Earth: Check your assumptions.
Air tells Fire: "Fools rush in where wise men never go…"
Air tells Water: Don't put me in a box.
Earth tells Air: Do you have any idea what everyone else thinks?
Earth tells Fire: You didn't care nearly so much about this today as you did yesterday.
Earth tells Water: Find out what's actually true (about yourself, or the world), and stick with that.
Fire tells Air: "...but wise men never fall in love"
Fire tells Earth: The world needs this, and if we wait we may never be able to do it.
Fire tells Water: "when we act, we create our own reality. And while you're studying that reality—judiciously, as you will—we'll act again, creating other new realities, which you can study too, and that's how things will sort out. We're history's actors…and you, all of you, will be left to just study what we do."
Water tells Air: Choose something for a while so the world knows who you are and what you want. You can change later if it doesn't work. Just give the world some information.
Water tells Earth: A fixed point looks different relative to different groups and people. Context is everything.
Water tells Fire: Be more responsible for how you affect the things around you.
Do you find yourself agreeing with some of these and passionately disagreeing with others? Do you find yourself saying them, or often said to you?
Lastly, in a comical sense I identify these with various superpowers.
Air: teleportation, omnipresence
Water: shapeshifting, healing
Earth: invulnerability, protection
Fire: laser beams, catalyzing
Update: I made a personality quiz based on this post: http://www.proprofs.com/quiz-scho
 Cowboy Bebop: Spike is Water, Jet is Earth, Faye is Fire, Edward is Air. http://en.wikipedia.org/wiki/List_of_Co
Avatar: the four nations have different personalities meant to reflect their element.
 As someone who identifies as Water, it was easiest for me to come up with the cons for this class. I do not mean the cons in this, or any of the categories, apply to anyone specific reading this. In fact, they are really more “words with negative connotations for the neutral traits otherwise described here”.
 Still thinking on two of the arguments, will fill them in later.