Elementalism is a somewhat old-fashioned term from various fantasy games and settings used to refer to a form of magic that focuses on the manipulation of the elements. Exactly what defines an "element" depends on the setting -- in addition to more iconic forms like fire, earth, air and water, some settings also cover more abstract forms, such as lightning, ice, metal, or wood. Some settings even go so far as to add in "esoteric/conceptual" elements -- things like Spirit, Death, Life, Gravity, Sun, Moon, etc.
This form of magic is most associated with videogame RPGs, as it readily translates into a handy array of attacking, buffing and debuffing spells, but it has made appearances in traditional /tg/ media as well.
Although elementalism refers to the practice of elemental magic as a whole, elementalists usually only practice one specific kind of elemental magic -- for example, a caster who uses nothing but fire -- or else a small group of "related" magics, such as a "wood elementalist" who uses plant-manipulating spells (the literal interpretation of wood), air-manipulating spells (because of wind's association with the wood element in the Wu Xing) and lightning spells (because of lightning's connection to air).
Elementalism usually involves a form of Rock-paper-scissors, which furthers the tendency for elementalists to only use a specific kind of elemental magic.
Elementalism in D&D
In Dungeons & Dragons, Elementalism was an alternate form of specialization for the Wizard in Advanced Dungeons & Dragons, via the Elementalist specialist, though it never really caught on, and also created the Wu jen as an attempt at an "Oriental Elementalist" spell-caster that is supposed to draw its powers from the Wu Xing elemental setup of Earth, Fire, Water, Wood and Metal.
In 4th edition, the Sorcerer received some elemental traits in the form of both a magical origin (the Storm Sorcerer, in Arcane Power) and the Essentials "Elementalist" Variant Class. The Warlock similarly possesses an Elemental Pact. Dragon Magazine also created a Pyromancy school for the Mage, which was a classic mono-element fire caster.
5e continues this by having no fewer than four elementalism-themed sorcerer subclasses; the Storm Sorcerer (originally the Waterborne Adventurers Unearthed Arcana, then Sword Coast Adventurer's Guide & Xanathar's GUide), and the Phoenix, Sea and Stone Sorcerers (all in the same Unearthed Arcana). Ironically, the best elementalist in 5e to date is the "Order of the Wu Jen" Mystic, who uses psionics rather than arcane magic.
Elementalism in Pathfinder
Needless to say, with its Archetype system, Pathfinder has readily embraced the use of Elementalism for the Wizard. There are no fewer than eight Elementalist archetypes, with another four representing "focused" elementalists. The "classic" quartet (Earth, Air, Water & Fire) appeared in the Advanced Player's Guide, whilst the "oriental" elements of Wood & Metal appeared in Ultimate Magic, with Void appearing in the Dragon Empires Primer. Finally, the Aether element and the focused elements of Ice, Smoke, Mud & Magma appeared in the Elemental Master's Handbook.
Of course, this is all overlooking the Kineticist class from Occult Adventures, a class that literally channels various elements into missiles and various other miscellaneous effects.
Elementalism in Warhammer
In Warhammer Fantasy, the conventional schools of magic are based in elementalism, though of the "esoteric" approach -- for example, there's a school of Fire Magic, but there's also a school of Life Magic (manipulation of earth, water and plants, plus healing), a school of Heavens Magic (manipulation of stars, wind and lightning, plus divination and blessing/cursing), and so on.