An Element is a part of a whole, and it is usually an irreducible, fundamental part. For instance, a circuit element, like a resistor or capacitor, cannot be broken up into a collection of other circuit elements, and an elementary particle is not made up of other particles (until we figure out how to split it, anyway -- the word "atom" comes from the Greek word for "indivisible," but obviously that didn't last).
Most systems of magic will include "elemental magic" (usually called Elementalism) of one kind or another, based on the notion that the universe is composed of a few elemental forces (exactly what these elements are depends on the setting), and that a person with the right combination of natural talent and training can wield these raw forces as tools and weapons. Some settings include elementals, creatures which are made of or otherwise embody one of the elements, and which may be summoned and bound by mages, or may be intelligent in their own right.
Once you open the door to elemental magic, you inevitably get to issues of balance. Often, the elements play a big game of rock-paper-scissors with each other, with each element being strong and/or weak against the others; for example, dousing a fire elemental or fire mage will leave him sputtering and steaming.
The list of chemical elements has 118 members in it and keeps growing as more atoms get smashed together, although only the first 94 exist in nature. This would not make for a very fun system of magic. The list of elementary particles isn't much better -- the standard model has 17 of them, and that's before we start getting into antiparticles, colors, gravity, string theory, and supersymmetry. There are only four fundamental forces (gravity, electromagnetism, the weak nuclear force, and the strong nuclear force), which is a manageable number, but they're ludicrously imbalanced -- it's easy to think of gravity as being the dominant force on planetary scales, but even a simple fridge magnet is strong enough to hold onto the wall with the whole 6.580 sextillion tons of Earth pulling down on it. EM is important from stellar down to atomic sizes, and the strong and weak nuclear forces only matter at the subatomic scale. Never mind the question of what a strong-nuclear-force elemental would look like.
The most familiar system of elements to Western audiences is probably the classical Greek system of fire, air, water, and earth. Almost all elemental magic systems include at least these four. If the elements are arranged in that order around a square, then they will naturally form pairs of opposing (and thus balancing) elements, as well as transitions between an neighboring elements, which might be manipulated by two elemental mages working together (for example: earth + water = mud) or fought over by two opposing elemental mages. Sometimes there will be a fifth element thrown into the mix, like "quintessence" (which is just Greek for "fifth element") or "heart" (if you're trying to summon Captain Planet).
These four elements match up to four natural states of mater: solid Earth, liquid Water, gaseous Air, plasma Fire.
The Chinese elements or "changes" are probably the other major system which inspires magic systems today. The five elements are, in order, Fire, Earth, Metal, Water, and Wood. Each element "creates" the next in the sequence, "destroys" the element two steps over, and "insults" the element three steps over. The Wu jen from 3e's Complete Arcane is a spellcaster based on these elements, complete with their own set of themed spells.
Settings that feel like doing things differently may use the Japanese elements of Earth, Water, Fire, Wind, and Void (or "spirit"). The Legend of the Five Rings uses them.
Magic: The Gathering
Magic: The Gathering doesn't really have "elements" in the sense that most people think of them, but the five colors of magic (white, blue, black, red, and green, arranged in a pentagon or wheel) do behave in a similar way. Each color's "allies" are its neighbors, while the colors opposite are "enemies". This concept is known as the Color Pie.
For more detailed information, see Type on Bulbapedia.
Pokémon (the creatures) and their attacks have elements, called "types". In the video games, the type of the attack is checked against the type of the defending Pokémon; if the attack is of an element to which the defender is weak, it is said to be "super effective!" and causes double damage, while if the defender has a resistance to the attack element, the attack is described as "not very effective..." and causes half damage. The card game works similarly, but it compares the type of the attacking Pokémon rather than the type of the attack itself, and rather than doubling or halving damage, damage is incremented or decremented.
The video games started out with the types of Normal, Fighting, Flying, Poison, Ground, Rock, Bug, Ghost, Fire, Water, Grass, Electric, Psychic, Ice, and Dragon. Generation II (Gold, Silver, and Crystal) added Steel and Dark, and Generation VI (X and Y) added Fairy. Pokémoncan be of two types at the same time.
The card games use a reduced set of types: Grass, Fire, Water, Lightning, Fighting, Psychic, and Colorless to begin with, and then Darkness, Metal, Dragon, and Fairy were added as the list of types grew in the video games.
In the Dungeons & Dragons Planescape campaign setting, there are Elemental Planes corresponding to each of the four classical elements, Paraelemental Planes at the boundaries between neighboring Elemental Planes (Smoke between Fire and Air, Ice between Air and Water, Ooze between Water and Earth, and Magma between Earth and Fire), and Quasielemental Planes at the boundaries between the Elemental Planes and the Positive and Negative Energy Planes:
Yes, some grid-filling psychoes have been crazy enough to figure out what the quasielemental/paraelemental planes (that is, the Positive and Negative Energy versions of Smoke, Ice, Ooze and Magma) are. It's never been done officially, though.
It is from these planes that magic-users draw to manifest most of their effects: healers pull from the Positive Material plane to treat injury and illness, wizards pull fire and lightning from the requisite planes, and dispelling magic or creating undead means pulling from the Negative Material. Most planes are inhabited by elemental creatures, though those of the "pure" planes tend to be stronger while those of the Paraelemental Planes are mostly weak mephits. Some elementals are intelligent and sapient, and, with the right protective spells for both parties, capable of interacting with and enjoying the company of the "muddled" creatures of the Prime Material planes. Yes, in that way too. A creature with elemental parentage is called a genasi.
The most powerful of all elemental creatures are the Archomentals, known in earlier editions as the Elemental Princes (or, more commonly, the Princes of Elemental Evil/Good) and in 4th edition as the Primordials.