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The simplest definition of convergence implies that $X_n$ will converge to $X$ if for any arbitrary number, say $\epsilon > 0$, $|X_n-X|< \epsilon$ for all $n>N(\epsilon)$ so that $\lim_{n \to \infty} X_n=X$. How can I interpret $N(\epsilon)$?

... and how I should have written the message above using MathJax?

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    $N(\varepsilon)$ is a natural number which may depend on $\varepsilon$. In other words, once $\varepsilon$ is chosen, you may pick $N$ depending on that $\varepsilon$. – 2017-01-10

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The intuition behind the definition of convergence of a sequence $\{X_n\}$ to $X$ is that, given some amount of error (i.e. given $\varepsilon>0$), we can go far enough in the sequence (i.e. looking at terms past $N(\varepsilon)$) and only get values whose error is less than $\varepsilon$ (i.e. $|X_n-X|<\varepsilon$ for $n\geq N(\varepsilon)$).

In other words, given $\varepsilon>0$, the integer $N(\varepsilon)$ tells us how far we have to go in the sequence to ensure that the error in replacing the limit by a value from the sequence is no greater than $\varepsilon$. How one determines $N(\varepsilon)$ depends on the sequence in question, and there is no general method for determining $N(\varepsilon)$.

And to answer your second question, the quantified definition of a limit, that is, $\forall\varepsilon>0\ \exists N(\varepsilon)\in\mathbb N\ \forall n\geq N(\varepsilon)\ |X_n-X|<\varepsilon$, should be typeset as 

$\forall\varepsilon>0\ \exists N(\varepsilon)\in\mathbb N\ \forall n\geq N(\varepsilon)\ |X_n-X|<\varepsilon$
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    As a side note, there are two different built-in epsilons: \epsilon (which gives $\epsilon$) and \varepsilon (which gives $\varepsilon$). I prefer to use the latter, as $\epsilon$ looks too much like $\in$. But the choice is yours. – 2017-01-10
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    As another side note, when looking at stack exchange on a PC, you can always right click on any mathematical expression and choose Show Math As, then TeX Commands to see what the input was. – 2017-01-10