Like other games, such as Monopoly, a wargame has a board (or map), items to facilitate play, (game pieces, cards, dice) and a set of rules to govern play. Like Monopoly, just as in any other game, you win by achieving the game's victory conditions. Unlike family games, wargames typically model conventional warfare (or political/revoltionary warfare etc.). Wargaming is an umbrella term.
Tactical wargames might have counters representing individuals or groups of combatants like squads or platoons (Advanced Squad Leader by MMP). An operational level game might have you commanding dozens of regiments (Three Days of Gettysburg by GMT Games), and a strategic level game would have you commanding one or more nations (Advanced Third Reich by MMP).
Some wargames have a map no larger than the cover of a magazine, use no more than a dozen counters, and maybe four pages of rules. A game like this can be played in twenty minutes or less. At the other end of the spectrum, the maps required by some war games would sprawl over an area much larger than a ping pong table (Europa by GRD), involve several thousand counters, have over 100 pages of rules, and require several months - at a minimum - to play.
There are "war games" on political upheaval (1989 by GMT) and revolution (The Russian Civil War by SPI) and terrorism (A Distant Plain by GMT Games). The U.S. fights the Soviet Union in a global proxy war (Cold War by SPI) in the last half of the 20th century. And (Civilization by Avalon Hill) requires you to take your culture through the epocs of history over a period of centuries.
Although a well designed wargame may give you some real insights into the subject matter it is modelling, and competitive game play is attractive to many, its social dimension cannot be understated. If you are like me, you wargame formost to socialize. For decades, I have sat down with close friends on a weekly basis, and gas-bagged on current events, sports, politics etc. You just need a game with a topic you're interested in, some good friends, and a little imagination. You'd be surprised how many of the world's problems we've solved on Tuesday nights.