Elemental (Monster)
An Elemental is a nebulous breed of monster defined by its strong connection to the power of the Elements. The most iconic depiction of the elemental is as the incarnate spirit of a given element, causing it to manifest as a (usually roughly humanoid) entity comprised solely of elemental matter - a swirling mass of wind or flame, a mobile clump of earth or a slime-like animate puddle of water. However, many different monsters have been considered "elementals" over the years.
Real-World
As a basic concept meaning "magical being that embodies or personifies a force of nature", the elemental has been around a very long time. The idea is practically universal, and has its roots in everything from gods and nymphs to more modernistic "fairy tale" beings like Jack Frost.
The most well-known real-world belief about elementals, and probably that which has had the greatest impact on fantasy gaming, is that created by the 16th century alchemist Paracelsus, who defined four main "races" of elemental - Gnomes of Earth, Salamanders of Fire, Sylphs of Air and Undines of Water - as part of his personal alchemical philosophy. Other prominent alchemists with similar theories include Heinrich Cornelius Agrippa and the Comte de Gabalis, whilst elementals were integral in the belief system of the Rosicrucians.
Dungeons & Dragons
In Dungeons & Dragons, elementals are Outsiders (what other settings might call "spirits") native to the Elemental Planes, and thus take the form of roughly humanoid creatures made up of elemental (or paraelemental, or quasielemental) matter. This has led to a vast and sprawling family of elementals and elemental-like creatures over the various editions, starting with Advanced Dungeons & Dragons.
Your standard Elementals are, well, as described; vaguely humanoid masses of elemental matter. They are the most populous of the various elemental beings.
Dread Elementals are standard Earth, Air, Water and Fire Elementals that were summoned to the Demiplane of Dread, Ravenloft, and were warped into horrible versions of themselves. Grave Elementals are warped Earth Elementals who are made of grave soil and have the ability to bury people alive with a touch. Mist Elementals are corrupted air elementals who have the ability to turn people Chaotic Evil with a touch. Blood Elementals are tainted Water Elementals who can drain the blood from others to feed themselves. Pyre Elementals are distorted Fire Elementals who have the ability to either burn through armor (AD&D) or animate corpses as burning zombies or skeletons (3e).
Genies in D&D are considered a breed of elemental, if very distinct from the standard masses. This is why there are four distinct breeds, one for each element; Dao of Earth, Djinn of Air, Efreet of Fire and Marid of Water. Plus the fifth "mixed elemental" Janni.
AD&D introduced the concept of "Elemental Kin"; entities that were very strongly related to the elementals, but looked and functioned more like humanoid races: Air was home to Aerial Servants and Sylphs; Earth was home to Chrysmals, Pech and Sandlings; Fire was home to Azers, Salamanders and Tome Guardians; Water was home to Nereids.
It also introduced the concept of Elemental Weirds. These were originally portrayed as serpentine Elemental Kin, with only Earth Weirds and Water Weirds existing, but in 3rd edition these were changed into their own unique Nymph-like elemental Oracles, with one for each plane. The original Elemental Weird became the Lesser Elemental Weird, a larval state, and didn't get acknowledged until Dragon Magazine #347.
Naturally, Ravenloft fans jumped in and added Dread Elemental-Kin in the Book of Shadows netbook; Myst Sylphs, Mist Servants, Crypt Gnomes (corrupted Pech), Gravelings, Heat Stalkers (corrupted Salamanders), Pyre Wyrms (corrupted "Fire Snakes" - larval Salamanders), Bloody Marys (corrupted Nereids) and Scarlet Streams (corrupted AD&D Water Weirds).
One creature unique to AD&D that didn't make it into subsequent editions was the "Animental"; an animal or monster (including monstrous humanoids, like medusas) who died and whose spirit was drawn into an ELemental Plane, where it was reincarnated as an elemental version of its former self. This idea probably inspired the Elemental Template in 3rd edition.
Grues are small, malevolent, elemental spirits, with one for each of the four Elemental Planes.
Al-Qadim introduced the Elemental Vermin; small animals native to the foru Elemental Planes and which basically serve as their home environment's form of pest. Theoretically, they could serve as familiars.
Archomentals are the powerful demigod-like rulers of the Elemental Planes.