Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. The fifth letter of the Greek alphabet. See Table at alphabet.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The name for the fifth letter of the Greek alphabet, ε or Ε, preceded by delta (Δ, δ) and followed by zeta (Ζ, ζ).
 n. In IPA, the phonetic symbol that represents the openmid front unrounded vowel; represented in SAMPA as E.
 n. An arbitrarily small quantity.
 n. A negligible effect.
from The Century Dictionary and Cyclopedia
 n. The fifth letter of the Greek alphabet, equivalent to short e.
 n. In mathematics, a quantity which approaches zero when the independent variable approaches a certain limit fixed for it by the conditions of the particular problem or discussion.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. the 5th letter of the Greek alphabet
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Let's look at the second number, epsilon, which is roughly .007.
Rabbi Shmuley Boteach: Does Questioning Evolution Make You AntiScience?

Hilbert developed a method for the study of consistency problems, called the epsilon substitution method, to deal with the quantifiers.

The EKG may show inverted t waves in right sided leads and the socalled epsilon waves.

I've been listening to a few odd bits and pieces of music these last few days (wanted to try and get the word epsilon into the last 3 words before the bracket, but couldn't ... conversation today re: possible other name for one of the rooms in MVB (Merchant Venturer's building, home of CS ppl ehre in Bristol) which is called 0.01, which spawned that idea floating around in my head).

If you carry two copies of a variant called epsilon 4, then your risk of succumbing to Alzheimer's disease is greatly increased.

The pressure dependence of the melting point is indeed the key to the solution  it turns the arbitrarily small "epsilon" temperature difference into a finite delta T, the difference between the freezing point of water under the pressure of the weight and the melting point of ice after the weight is removed.

For any finite "epsilon", the infrared problems are absent.

The measurable results are independent of "epsilon".

You should ask what is the cross section to create two photons plus an arbitrary number of photons whose energy is so small, below "epsilon", that you can't really detect them.

Once you admit that you can't measure the photons whose energy is below "epsilon", you eliminate the infrared divergences from the loop diagrams, and the part that diverges for
qroqqa commented on the word epsilon
By that Friday before the IMF meeting, we were literally one epsilon away from a systemic financial meltdown.
—talk by economist Nouriel Roubini, 30/09/2008
This is the first realworld use of the jargon sense of 'epsilon' I've seen.
November 20, 2008