Please explain the definition of cyclic groups etc.

Perhaps a few examples will clarify. Let $S = \{a,b,c\} \subseteq G$.

$abac$ is a finite composition of elements of $S$. Here, "composition" is just another word for the group operation, maybe you also call it "multiplication. Finite means exactly what you think it does. This is a finite composition because it is a composition of finite things. Other examples include $a$, $bbb$, $abcbabbaca$.

We're not just using elements of $S$, we're including "...elements in $S$ or their inverses." So we also allow $\{a^{-1},b^{-1},c^{-1}\}$ in our finite compositions. So now we are allowed for instance $ab^{-1}cb^{-1}abb^{-1}aca^{-1}$.

Thus, $S \subseteq G$ generates $G$ if every element of $G$ can be written in the way just described.

A special case of this is when $S = \{a\}$, in which case $G$ is just all powers of $a$: $G = \{\ldots, a^{-2},a^{-1},e=a^0,a^1,a^2,\ldots\}$ (note that $a$ might have finite order, so $G$ might still be finite.) In this case, we say $G$ is cyclic.

Edit: As mentioned by a comment, "$a$ has finite order" means that there is some $n \in \mathbb{N}$ so that $a^n=e$. In particular, this mean $a^{n+1}=a$. Then $G$ (from above) can be written as

\begin{align} G &= \{\ldots,a^{-2}, a^{-1},e=a^0,a^1,a^2,\ldots, a^{n-1},a^n=e,a^{n+1}=a,\ldots\} \\ &=\{\ldots,a^{-2},a^{-1},e,a,a^2,\ldots,a^{n-2},a^{n-1},e,a,a^2,\ldots \} \end{align}

After $n$ elements, the list repeats, or "cycles." Because sets don't see repitition, we may rewrite as

$$G = \{e,a,a^2,\ldots, a^{n-2},a^{n-1}\}$$

Hence, $G$ is finite.