Meta happens when we go from talking about X, to talking about talking about X. Some examples:

• Wikipedia's example: "This debate isn't going anywhere." If we've been talking about politics, now we're talking about our discussion of politics.
• A self-hosting compiler is a compiler for language X that is written in language X. So it parses, optimizes, and "understands" programs written in X; and can be run on itself.
• This is not a pipe.
• A meta-joke is a joke about jokes, or one whose humor relies on the hearer's expectation that a particular kind of joke is being told, with the punch line revealing that it's not that sort of joke at all.

"Meta" is also used to mean "at a higher level of organization". For instance, if there is a competitive game G, then the metagame of G is the interaction of different styles of play in a given pool of players.