A wargame is a game that depicts a military action. Wargames cover a wide range of topics, from ancient times to the present or even the future, and can cover everything from actions between small units on a very small board to a world war on large maps that depict the entire globe - or very large maps depicting one battle at a very fine level of detail. Although most wargames are based on historical battles or wars, there are wargames based on fantasy or science fiction, as well as wargames based on hypothetical but historically-based situations (for example, a war between the West and the Soviet Union in 1948). Probably the most popular period for wargames is World War II, followed by the Napoleonic Wars and the American Civil War. However, wargames cover a vast range of conflicts, and if you are interested in a particular war you are quite likely to be able to find a game that covers it in some way.
A wargame will be at one of three levels of warfare - tactical, operational, and strategic. Tactical games depict a skirmish, battle or series of battles, using smaller scale units and maps that depict a battleground that is a few miles or less - sometimes much less - in size. Examples of tactical wargames include Advanced Squad Leader and Combat Commander: Pacific.
Operational wargames cover a broader scope of military actions, greater than single battles. These may cover an entire smaller war, or a series of operations or a campaign within a greater war. Units are larger than in tactical games. Examples of operational level wargames include Fortress Europa and 1914: Twilight in the East.
Wargames on the strategic level usually recreate a major war on a large scale. Typically units in such a game will be corps or army level, although in so-called "monster" wargames (no set definition, but usually a game with large maps and lots of counters that takes a long time to play) divisions and even smaller sized units can be depicted. Strategic wargames are more likely than tactical or operational level games to include big-picture issues such as production of new military units. Examples of strategic level wargames include World in Flames, Paths of Glory, and Here I Stand.
Beyond the level of wargame are other choices - for example, games that focus exclusively on naval battles and campaigns are common. Examples include Wooden Ships & Iron Men and Sixth Fleet. Similarly, wars depicting air battles are popular.
Within these rough definitions are a huge variety of mechanics, themes, and approaches, and many levels of complexity; it is generally accepted that on average, wargames are more complex than other types of games, although this is not a hard and fast rule. Naturally, all wargames include combat - although not all games that include combat would typically be deemed wargames; for example, Imperial and Age of Empires III: The Age of Discovery have combat and military forces, but most wargamers would not see them as wargames. Some wargames include political considerations, for example Ici C'est La France or Triumph of Chaos.
Most wargames are designed for two players, particularly tactical games. However, a significant minority are designed to be playable by three or more. Examples of wargames for more than three players include Crown of Roses, World in Flames, Friedrich, Here I Stand, and Pax Romana.
Victory conditions in wargames vary greatly, as does the subject matter. Common ways to win wargames include capturing one or more objectives on the map; destruction of a certain portion of an enemy force; accumulating victory points gained by a variety of means including taking objectives and destroying enemy units.
Most wargames use cardboard counters to depict military units, with symbols and numbers on the counters that describe the units' attributes. Common attributes include the unit's size, type (i.e., cavalry, cruiser, archer, mechanized infantry...) combat strength, and speed of movement - but many other attributes such as morale, status of the unit, nationality, etc can also be found. Wargames that use miniature figures to depict military units are also quite popular. So are wargames that use blocks, which hide information on a unit from the opponent.
Finally, there are many different approaches to wargame mechanics. For movement, mechanics include hex-based movement (example: The Russian Campaign), area movement (Warriors of God), or point-to-point movement (Friedrich); note that is is rare for movement to rely on the roll of dice. Combat can be resolved by rolling dice and referring to a combat resolution table (CRT) (example: The Rise and Decline of the Third Reich or Lee Vs. Grant), or by rolling dice to "hit" opposing units (example: Wellington or Warriors of God), or by computing odds and drawing a card to see the result (Kingmaker). A popular mechanic over the past 15 or so years is the card-driven game (CDG), where cards are played either to activate units for movement or for the benefits described on the card such as reinforcements; We the People is seen as the first such CDG, and other examples include Paths of Glory, Here I Stand, and Empire of the Sun.