The hyperreal numbers are numbers which extend the usual real numbers. The idea is that in the real number, if $x>0$ then for some $n\in\Bbb N$ we have that $\frac1n<x$.
In the hyperreal numbers we don't have that. There are numbers which are smaller than any $\frac1n$, but are larger than $0$. Note that these numbers are not real numbers, but that's okay since the hyperreal numbers extend those.
However, the hyperreal numbers and the real numbers share many properties, both are fields and basic statements we can write down in the language of fields (namely the existence of roots of certain polynomials) are true in the real numbers if and only if they are true in the hyperreal numbers. These similarities can be extended, but that requires some additional knowledge in mathematics.
Two important differences (except the existence of infinitesimals) are these:
The hyperreal numbers are not necessarily unique. Namely, there is exactly one field (up to isomorphism) which is $\Bbb Q$, and there is exactly one field which is $\Bbb R$ (again, up to isomorphism). However the statement that every two hyperreal fields are isomorphic requires additional assumptions on the mathematical universe.
The hyperreal numbers do not make a complete field. Namely, in $\Bbb R$ is a set is bounded, it has a least upper bound. In the hyperreal numbers this is not true anymore. Consider $\Bbb N$, then since there is $x>0$ such that $x<\frac1n$ for all $n\in\Bbb N$, it follows that $\frac1x$ is an upper bound for $\Bbb N$, but in that case $\frac1x-1$ is a smaller upper bound. So there is no least upper bound.