"the world's mine oyster..."
See in text (Act II - Act II, Scene 2)
Today if we use this expression, we are making a claim that the world's riches are ours for the taking. However, the ensign Pistol here uses the phrase more as a threat. Since Falstaff refuses to give Pistol any money, Pistol says he'll use his sword to pry money out of other helpless victims. The thievish intentions behind this phrase have mostly been forgotten, having become a vain pronouncement of opportunity.
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"what the dickens..."
See in text (Act III - Act III, Scene 2)
This oath or expletive substitutes "dickens" for "devil" much in the same way other words like "gosh" and "marry" are substituted. This softens the oath and also serves to make it have a more comical effect, particularly in cases like, "What the dickens!"
"I have a kind of alacrity in sinking..."
See in text (Act III - Act III, Scene 5)
In this scene, Falstaff has returned from being previously thrown out with the dirty laundry to avoid a compromising situation with Mrs. Ford. He complains about how humiliating being tossed in the river was, and how due to his enormous size and weight, he has an "alacrity in sinking." "Alacrity" means a kind of brisk readiness, so the joke is that he is so heavy that he would have sunk straight to hell if the river had been deep enough.