I've been tossing the word "Mechanic" around a lot without ever giving it a proper definition. I plan to rectify that situation now.
A mechanic is any rule or set of rules that define the way a game is played.
Not at all vague, is it? While it's terribly easy to point to something that's a mechanic, it's much harder to make a general definition that encapsulates them all. So let's whip out a couple examples of mechanics.
The motion of rooks, bishops and queens in chess.
The motion of knights, or even just the fact that they can jump other pieces.
The fact that pawns can't go backwards, or pawn promotion at the other end of the board.
In fact, the motion of pawns gives us an excellent example. Y'ever hear of the en passant? Lemme tell you the story of it's origins, (as I heard it, and not necessarily true). Nowadays the first move a pawn can make is either one or two spaces forwards. It used to be that pawns could only move one space forward. (Ignoring capturing rules.) Trouble is, this mechanic was unnecessarily delaying the game; at the start of the game a number of turns would pass that progressed the game very little. So people instituted the first pawn movement double jump mechanic. Yeah, I did work to make that name that awkward. The trouble is, that mostly cosmetic (cosmetic here meaning that it makes the game smoother to play, it doesn't have much effect on the strategy) change had a significant effect on the strategy, if only to contradict my parenthetical definition. If you'll recall, pawns capture only in spaces one diagonally forward from it's current position. In the olden days you could park a pawn in the fifth row (that is to say, two spaces in front of your opponent's pawns) and be confident that the pawns on either side couldn't sneak past. But with the new rule, if the moved two spaces on the first turn they could effectively bypass your blockade. So they invented the en passant, a new mechanic that basically lets the fifth row pawns to capture regardless.
Got that? Only one space forward was a mechanic. Two spaces forward is another mechanic, designed to move the early game along. En passant is yet another mechanic, designed to neutralize the strategic implications of the two spaces forward mechanic.
Beyond specific rules and applications of the rules, the word "mechanic" can describe large parts of a game's structure. For example:
Going 'round the board, landing on spaces in monopoly is a mechanic.
So is the division of movement into combat and reinforcement in RISK
The differentiation of costs by unit type in Axis & Allies qualifies.
Experience and leveling up in Dungeons and Dragons also qualify.
So what use is it to define something that's so, well, universal? You have to note that whenever you're talking about a particular mechanic you're always making a statement about that mechanic. You're not just defining it, you're explaining why that mechanic exists, what it does, how it affects game play, or something about it.
Going 'round the board in monopoly works as a method of randomization of results. (chance cards, $200 for the railroad, that friggen hotel on boardwalk; what's this? Mediterranean avenue is still for sale? Probably cause it's a dump)
The zombie killing mechanic in ZOMBIES!!! allows players to kill zombies, so it's good they put it in. Said players wouldn't hang around very long if you sold 'em a game called ZOMBIES!!! that didn't allow them to kill zombies.
The mechanic of using play money to determine victory points tends to work out very well in financial games, not only does it make sense from a realism perspective but psychologically it's very easy to get players to care about those same piles of play money.
As a matter of fact, mechanics are introduced into games for one (or both) of two reasons. Either they make the game play better, or they increase realism.
In RISK, getting armies by turning in cards is a mechanic that's largely for game play reasons. If the mechanic didn't exist, players would exhaust their supplies of armies and be reduced to squabbling over borders, possibly with no end in sight. (I've never actually played the game sans cards to test this.) There's very little real world reason why it would work this way. (America enters the war! Oh wait, America was already part of the war? Nevermind.) The game solar quest had a strictly realism based mechanic. In it, you had to be moving quickly enough to get from one planet's orbit to another. While this might make sense from an astronomical perspective, in gameplay terms it usually meant you were stuck orbiting Jupiter using up your fuel and having to pay rent to the jerk who already owns the moons and you wouldn't believe his prices.
It's relatively common to find mechanics that both make sense from a realism perspective and work to improve the game play. For example, in Monopoly, the more you build up a property the more it costs people who land there. And owning a monopoly on a particular type of properties increases the amount you can charge. It makes sense from a real world understanding of what a monopoly does, and it also provides a valuable game function (It bankrupts your eight year old cousin so that, even though he's going off in tears you can at least declare this stupid game to be over.) Similarily, the different prices of units in Axis and Allies serves the realism goal because honestly, a submarine is more expensive than a tank. It also makes for a better play experience by forcing people to decide how much quality versus quality they want in their forces. (Mein Fuhrer! I have... unfortunate news about your plan to build an army solely out of land battleships.)
One last point. When I build a game from scratch, I make the framework out of mechanics that I think would be interesting together. It's a world war, set in Asia, only you have to buy your oil on the free market. Steal the war mechanic from Axis & Allies, steal the free market mechanic from Power Grid, see how well they work together. Or take the Terrible Secret of Space. The very first thing I thought of, even before I had "space war" as the creative framework, I knew I wanted a game that mimicked the in depth strategy of Chess by Mail. I'm also looking for ways to expand the Dungeon Master mechanic out of your classic RPGs. Put two and two together and you've got a new kind of internet gaming.
I hope you've got a better idea now of what the hell I'm talking about. Every little bit helps.