An example of second conjugate we know that $X$ is a normed vector space and $X'$ is the conjugate of $X$ (set of linear bounded functional defined on $X$) and $X''$ is the second conjugate of $X$ (i.e. the set of of linear bounded functional on $X'$).
As example... $X$ could be set of continous function on a compact interval $I$ while an element of $X'$ could be an integral operator what could be an example of instance belonging to $X''$?
 A: The main objects in $X''$ are the evaluation maps: $(T_x)(f)=f(x)$ where $x \in X$. (My $T$ is called the canonical embedding of $X$ into $X''$.) If $X$ is reflexive (which is true in many important examples, such as $L^p$ spaces for $1<p<\infty$), then this is everything in $X''$. 
Otherwise $X''$ contains additional objects. It may or may not be easy to see what these objects look like. @TomekKanla's answer deals with the case $X=C[0,1]$. A nastier case is the case $X=L^1([0,1])$. Here, under the axiom of choice it is possible to prove that there exist members of $(L^1)''=(L^\infty)'$ which are not given by integration against an $L^1$ function. Instead they are given by integration against so-called "finitely additive measures". In the context of the Lebesgue measure, such a thing has never been constructed, and probably can never be constructed.
A: Let $X=C[0,1]$. Then any bounded Borel function $f\colon [0,1]\to \mathbb{C}$ defines an element of $C[0,1]^{**}$ via
$$\langle f, \mu \rangle = \int\limits_{[0,1]} f(t)\,\mu({\rm d }t)\qquad (\mu \in C[0,1]^*).$$
where we identify $C[0,1]^*$ with the space of all regular Borel measures on $[0,1]$ using the Riesz representation theorem.
However the second dual of $C[0,1]$ is much larger than just the set of bounded Borel functions. It can be identified with the space $L_\infty(\mu)$ for some silly measure $\mu$ that is not even $\sigma$-finite.
(I write $X^*$ instead of $X^\prime$.)
