Question: This letter "$\varepsilon$" is called epsilon right ? What does it signify in mathematics ?

Answer: 
The greek letter epsilon, written $\epsilon$ or $\varepsilon$, is just another variable, like $x$, $n$ or $T$.

Conventionally it's used to denote a small quantity, like an error, or perhaps a term which will be taken to zero in some limit.

It's possible that you are confusing it with the set membership symbol $\in$, which is something different. When you see $x\in X$ it means that $X$ is a set, and $x$ is a member of the set. For example,

$$1\in \{1,2,3\}$$

is true, but

$$4\in\{1,2,3\}$$

is false.
[Hilbert's epsilon-calculus](http://en.wikipedia.org/wiki/Epsilon_calculus) used the letter $\varepsilon$ to denote a value satisfying a predicate. If $\phi(x)$ is any property, then $\varepsilon x. \phi(x)$ is a term $t$ such that $\phi(t)$ is true, if such $t$ exists. One can define the usual existential and universal quantifiers $\exists$ and $\forall$ in terms of the $\varepsilon$ quantifier:

$$\begin{eqnarray}
\def\hil#1{#1(\varepsilon x. #1(x))}
\exists x.\phi(x) & \equiv & \hil{\phi}\\
\forall x.\phi(x) & \equiv & \phi(\varepsilon x.\lnot\phi(x))
\end{eqnarray}
$$