In general, if $R$ is a ring and $S\subseteq R$ is a subset then the ideal generated by $S$ is defined as the intersection of all ideals which contain $S$ as a subset. In other words this is the smallest ideal (by inclusion) which contains $S$. The notation is $(S)$.
In particular, if $S=\{a\}$ is a set with one element then $(a)$ is the smallest ideal of $R$ which contains the element $a$. Now, if $R$ is a ring with unit then it is easy to see that the set:
$I=\{\sum_{i=1}^n r_ias_i: n\in\mathbb{N}\cup\{0\},\ r_i,\ s_i\in R\}$
Is an ideal which contains $a$. Hence $(a)\subseteq I$. On the other hand, $(a)$ itself is an ideal which contains $a$, thus by the properties of an ideal it must contain all the elements of $I$. So $I\subseteq (a)$ is also true. So this implies $I=(a)$, we found a relatively easy way to describe $(a)$.
If we also knew that $R$ is commutative then it would be even easier, then we would simply have $(a)=\{ra: r\in R\}$. (check it)
Alright, now to the definitions. An ideal $I$ is called principal if there is some $a\in I$ such that $I=(a)$. A ring $R$ is called a principal ring if all its ideals are principal.
