4X games

4X games are an expanding family of games that are linked through 4 categories of gameplay that are all crucial elements to the game.
eXplore, eXpand, eXploit, & eXterminate
User toerck summarized the four elements that make a 4X game here. A slightly refined and tighter definition is:

• eXplore the map with units, revealing the territory's terrain or something significant hidden within the territory; what is revealed will affect the game now and/or later.

• eXpand the size of the owned area by claiming territories on the map; gaining or losing territories will affect the game.

• eXploit the owned territories by improving them, gaining something, and using it to increase your units and/or power; those 3 things--improve, gain, & use--can happen in any order.

• eXterminate other players from the game using units that can be lost; it does not have to be likely that a player can be eliminated from the game, just possible.

If it is impossible to eliminate a player from the game, but everything else (including combat) is just like a 4X, the game is known as a 3.5X.

The four Xes do not have to happen separately, they can be folded together and happen at the same time. However, all four Xes will happen in a 4X game.

Since all territory is eventually claimed, attacking and/or eliminating a rival's presence can be the only way to achieve further expansion.