Definition of Principal ideal:
Let $R$ be a commutative ring with unity and let $a \in R$ . The set $\langle a\rangle = \{ra\mid r \in R\}$ is an ideal of $R$ called the principal ideal generated by a.
Doubt
What is the principal ideal in case of a subset of $R[x]$ (set of all polynomials with real coefficients) with constant term zero.
$A=\langle x\rangle $ will work but I actually don't understand what $\langle x\rangle$ mean here. I think
$$A=\{f(x) \in R[x]\mid f(0)=0\}.$$
How all these things fit together?
$\langle x\rangle$ is the set of all polynomials that can be written as a multiple of the polynomial $f(x)=x$. Let's take a look at your example:
Every polynomial $f\in R[x]$ with $f(0)=0$ can be written as $f(x)=g(x)x$ for some $g\in R[x]$. Vice versa, every polynomial of the form $f(x)=g(x)x$ satisfies $f(0)=0$. Therefore $\langle x\rangle = \{f\in R[x]\;|\;f(0)=0)\}$.