"A mathematician is a device for turning coffee into theorems" (P. Erdos)
Addendum: American coffee is good for lemmas.

Old mathematicians never die; they just lose some of their functions.

A topologist is a person who doesn't know the difference between a coffee cup and a doughnut.

A mathematician and an engineer are on desert island. They find two palm trees with one coconut each. The engineer climbs up the first tree, gets the coconut, and eats it. The mathematician climbs up the second tree, gets the coconut, climbs down, walks over to the first tree, climbs it, and puts the coconut up there. "What did you do that for?" ask the engineer. The mathematician replies, "Now we've reduced it to a problem we know how to solve."

A Neanderthal child rode to school with a boy from Hamilton. When his mother found out she said, "What did I tell you? If you commute with a Hamiltonian you'll never evolve!"

What's yellow and equivalent to the Axiom of Choice? Zorn's Lemon.

What's purple and commutes? An abelian grape.

What's lavender and commutes? An abelian semigrape.

What's green and homeomorphic to the open unit interval? The real lime.

What is yellow and is also a complete normed linear space? A Bananach space.

What is purple, commutes, and is worshipped occasionally? A finitely venerated abelian grape.

The cherry theorem (a puzzle that reminds some of calculus theorems)
Q: What is a small, red, round thing that has a cherry pit inside?
A: A cherry.

What's a polar bear? A rectangular bear after a coordinate transform.

Dictionary of Definitions of Terms Commonly Used in Math Lectures:
The following is a guide to terms which are commonly used but rarely defined. In the search for proper definitions for these terms we found no authoritative, nor even recognized, source. Thus, we followed the advice of mathematicians handed down from time immortal: "Wing It."

CLEARLY: I don't want to write down all the "in- between" steps.

TRIVIAL: If I have to show you how to do this, you're in the wrong class.

OBVIOUSLY: I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it.

RECALL: I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test...

WLOG (Without Loss Of Generality): I'm not about to do all the possible cases, so I'll do one and let you figure out the rest.

IT CAN EASILY BE SHOWN: Even you, in your finite wisdom, should be able to prove this without me holding your hand.

CHECK or CHECK FOR YOURSELF: This is the boring part of the proof, so you can do it on your own time.

SKETCH OF A PROOF: I couldn't verify all the details, so I'll break it down into the parts I couldn't prove.

HINT: The hardest of several possible ways to do a proof.

BRUTE FORCE (AND IGNORANCE): Four special cases, three counting arguments, two long inductions, and a partridge in a pear tree.

SOFT PROOF: One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.

ELEGANT PROOF: Requires no previous knowledge of the subject matter and is less than ten lines long.

SIMILARLY: At least one line of the proof of this case is the same as before.

CANONICAL FORM: 4 out of 5 mathematicians surveyed recommended this as the final form for their students who choose to finish.

TFAE (The Following Are Equivalent): If I say this it means that, and if I say that it means the other thing, and if I say the other thing...

BY A PREVIOUS THEOREM: I don't remember how it goes (come to think of it I'm not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows.

TWO LINE PROOF: I'll leave out everything but the conclusion, you can't question 'em if you can't see 'em.

BRIEFLY: I'm running out of time, so I'll just write and talk faster.

LET'S TALK THROUGH IT: I don't want to write it on the board lest I make a mistake.

PROCEED FORMALLY: Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses).

QUANTIFY: I can't find anything wrong with your proof except that it won't work if x is a moon of Jupiter (Popular in applied math courses).

PROOF OMITTED: Trust me, It's true.

If you've read this far, you've got issues.