The famous western system revolves around the classic four elements, usually divided into "hot"/"cold" (fire and air = hot, water and earth = cold) and "dry"/"wet" (earth and fire = dry, water and air = wet) axisnote In the original Greek thought, it was pretty much a metaphor for separation versus unity (heat spreading things apart, cold uniting them) and the form cohesion (dry being a distinct shape, wet being flexible/absent shape), but expect no-one to remember this. May be aligned with seasons (nowadays Air/Spring, Fire/Summer, Water/Autumn, Earth/Winter, but apparently originally starting in Greece as Water/Spring, Air/Summer, Fire/Autumn and Earth/Winter), and many other things.
The elements were also arranged in a system of "tiers", which could relate both to purity and position and was often used to explain how the universe at large was structured: essentially, the assumption was that the elements were naturally arranged in spheres around each other, and that it was in their natures to seek to move above lower elements and sink below higher ones — so, heavy and dense Earth sought the center, then came Water to form seas and lakes over Earth, then Air, then Fire to form the stars and Sun.
Aristotle, also believed that each element had a preferred direction to move. Thus in Aristotelian physics a rock falls because it has lots of Earth in it and Earth's "natural motion" is down, while a flame rises because Fire's "natural motion" is up.
Of course the ancient Greeks knew that whatever the heck the planets were doing they weren't moving in straight lines so Aristotle proposed they were made of a fifth element whose "natural motion" was circular. According to Aristotle, this was Aether, a weightless, colourless substance that formed the "celestial spheres" and caused moon, planets and stars to move. This was something of a departure from earlier descriptions — in Homer's times, Aether was considered to be "purer air", being what gods at Mount Olympus breathed (which suggests that gods need to breathe, which of course begs the question of how Poseidon can be the king of the seas), and generally, it was thought to be simply air that was higher than "regular" air humans breathed, filling interstellar space.
This is one of the main reasons Astronomy made no progress for hundreds of years. Thanks to Aristotle's brainwave people didn't bother asking why the planets were acting as they did because they had been educated to think that they were made of something so different that Earth's physics just didn't apply to them. Of course it's a bit harsh to put all the blame for this on Aristotle, it was his successors that (in keeping with scholarly thought of the time) simply assumed he'd got it right.
In Greek medicine every element had its associated humour and bodily fluid from the Four-Temperament Ensemble. Air was blood, as both were hot and wet, and the Sanguine temperament; cold and dry Earth was associated with black bile and thus Melancholia; Fire, being hot and dry, was element of yellow bile and Choleric Temperament, while cold and wet Water was associated with phlegm and Phlegmatic humour.
Hinduism and Buddhism
In Hindu and Buddhist traditions, the elements, or "tattvas", are more like states of being than anything else, forming a ladder from the solid earth to the heavenly light. They're aligned with the senses (earth = smell, water = taste, fire = sight, air = touch and "akasha" note Originally meaning "sky", it's now pretty much synonymous with "sound". It's always described as a pure nectar = sound. They're also aligned with the chakras, which represent their ladder: Muladhara at the bottom, Swadhisthana in the genitals, Manipura in the stomach/solar plexus, Anahata in the heart, Vishudda in the throat and Ajna in the forehead.
Chinese Tradition
In Chinese thought, the elements are Wood (which is also synonymous with Wind), Fire, Earth, Metal (which also includes Electricity and Light) and Water. They are pretty much described as states of matter and natural phases, so logically they're also closely tied with seasons (Wood/Spring, Fire/Summer, Metal/Autumn, Water/Winter, and Earth being the time between seasons) and life stages (Wood = birth/childhood, Fire = teenagehood, Earth = adulthood, Metal = old age and Water = old age or death). They're divided in the Yin/Yang axis (Wood and Fire = Yang, Metal and Water = Yin and Earth is both Yin and Yang). The Four Gods each represent an element (the often forgotten Yellow Dragon is the fifth, representing the Earth). The elements are also associated with personality traits, some good, some bad (fire is politeness and aggression, water contemplation and indecision, earth stubbornness and greed, wind flexibility and yielding, metal righteous and headstrong)note Not a comprehensive list and are sometimes used as a form of horoscope. The system of thinking also includes Elemental Rock-Paper-Scissors, with each element "nurturing" another at their own expense, being nurtured by a second, and suppressing or being suppressed by the third and fourth (for instance, wood feeds fire, which produces earth, melts metal, and is extinguished by water), although excesses of any given element can throw this system out of whack (eg: excess fire scorches earth and boils water).
In addition, Ba Gua further splits the elements, with Metal being divided into Heaven and Lake, Wood/Air into Wind and Thunder, Earth into Earth and Mountain, and Fire and Water remaining the same. Some correspondences are switched (i.e. Earth becomes pure Yin and Metal pure Yang, with Wood/Air being the "in between" element).
in Greece as Water/Spring, Air/Summer, Fire/Autumn and Earth/Winter), and many other things.