Why epsilon is hyper-real number?
Is it defined as Infinitesimal?
What are the differences between infinitesimal to epsilon?
If you are not studying non-standard analysis this answer is going to make little sense.
In model theory systems can be enlarged. The standard system of real numbers $\mathbb{R}$ can be enlarged to $\mathbb{R}^*$ by the addition of one symbol $\epsilon$ having the property that $(\forall x\in\mathbb{R}^+)[0<\epsilon<x]$ The number $\epsilon$ is said to be infinitesimal while $\epsilon^{-1}$ is said to be infinite.
But to understand that answer you must study a full course.
SEE: Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 1.
The chapters and whole book is a free down-load at H. Jerome Keisler Home Page.
In most calculus courses $\epsilon$ is neither a hyper-real nor infinitesimal number. It is just an arbitrary positive number as in "Suppose $\epsilon > 0$". Such arguments then proceed to show some quantity $q$ satisfies $0 \le q < \epsilon$ and then conclude $q=0$. That idea is the basis of many calculus arguments and is frequently a stumbling block for students to grasp.