When Shaq accuses Dikembe Mutombo of “flopping,” what does he mean?
Shaq meant that when he posts up on Dikembe, catches the basketball, and then makes an agressive move towards the basket, that Dikembe falls to the floor INTENTIONALLY to draw an offensive foul call from the official.
In other words, Dikembe is feigning the receipt of a jarring body shot from Shaq for the sole purpose of getting a foul called on Shaq. Shaq sees it like this – Dikembe Mutumbo was well capable of staying on his feet & defending him, but Dikembe chose to “flop” instead.
Surely something analogous occurs in soccer–perhaps when an offensive player purposely takes a dramatic dive “for show” in response to a less-than-vicious tackle, in an effort to draw a card from the refs against the defender. Maybe even a penalty kick.
If there’s a sport where penalties or fouls are called, there’s flopping, with the possible exception of water polo, where the tactic is not very effective as you would have to go underwater I suppose. 
In hockey, it’s called diving. The NHL is cracking down on it, but in it’s typical weird way. Player A may trip Player B. B dives to the ice to make sure that the ref saw the play. The ref then gives A 2 minutes for tripping and B 2 mintues for unsportsmanlike conduct.
In football, flopping only really happens with kickers, who make sure to fall to the ground anytime they are hit after a kick in an attempt to draw a roughing penalty. Other than that, it’s pretty hard to draw a penalty in football by flopping since knocking someone over is a valued skill.
Five other American football flopping scenarios:
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QBs can flop after receving a glancing blow from a too-late pass rusher to draw a roughing-the-passer flag.
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Pass receivers can flop in an attempt to draw interference penalties.
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A defensive player (normally a lineman or linebacker), under oppurtune conditions, can “flop” to draw a holding penalty. This kind of flop needs an added gimmick–the defender hangs on surreptitiously to the blocker’s jersey.
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Rarely, a ballcarrier can flop when clearly running out of bounds and is gingerly shoved by a defender. The aim is to draw an unnecessary roughness (or personal foul) penalty.
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Rarest of all, a ballcarrier, in a particularly unusual situation, can “flop” when a defender’s hand is near their helmet or shoulder-pads area. The ballcarrier can jerk their head in the direction of any light impetus they may feel, and thus draw a heavier facemasking penalty (a 15-yarder instead of a 5-yarder or no call).
The NFL does have provisions for giving out penalties if the official feels that a kicker has “flopped”.
But if you want to see some high level flopping, I would say watch a soccer match.
One second, a player is writhing on the ground and a stretcher is brought out. A trainer wrings out a sponge on the guy and then he bounces back up ready to go.
And really, an offensive foul could probably be legitimately called on Shaq every time he touches the ball.
I used to think Mutombo was a pretty big guy. But now, every time I see O’Neal and Mutombo side-by-side all I can think of is a bowling pin and a ball–a medicine ball. So when they touch, Mutombo goes down sometimes. Big surprise.
At 7’2", Mutombo is half an inch taller than Shaq, but Shaq weighs about fifty pounds more.