What the epsilon symbol means
Epsilon (ε, capital Ε) is the fifth letter of the Greek alphabet. In math, ε stands for a very small positive number, as in the classic definition of a limit: "for every ε > 0 there is a δ > 0". Pair it with the delta symbol when you write limit proofs.
In physics, ε₀ is the permittivity of free space, about 8.854 × 10⁻¹² F/m, and ε also stands for strain in engineering and emissivity in heat transfer. In statistics, ε is the error term of a model such as y = β₀ + β₁x + ε. In computer science, ε is the empty string, and machine epsilon is the gap between 1 and the next larger number a computer can store as a floating-point value.
The euro sign € was designed with epsilon in mind: the European Commission describes it as a Greek epsilon crossed by two parallel lines.
ε vs ϵ, and why ∈ is different
Small epsilon has two shapes. ε (U+03B5) is the rounded form used in Greek text. ϵ (U+03F5), the lunate epsilon, looks like a half moon with a bar and is popular in math. LaTeX does the opposite of what most people expect: \epsilon gives ϵ and \varepsilon gives ε.
The set symbol ∈ ("is an element of") looks like a lunate epsilon but is a separate math operator, so write x ∈ A, not x ϵ A. ɛ is the Latin open e from the phonetic alphabet and several African languages, and the capital Ε looks identical to the Latin E but is a different character.
How to type ε
On Windows, hold Alt and type 238 on the number pad for ε. In Word, type 03B5 (or 03F5 for ϵ) and press Alt + X; in an equation, \epsilon and \varepsilon work too. On a Mac, use the Character Viewer (Control + Command + Space) or the Greek keyboard, where E types ε.
In HTML, use ε for ε and ∈ for ∈. For other set and logic signs, see math symbols; the full alphabet is on Greek letters.