What is the meaning of $\Bbb{Q}[x]/f(x)$? I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$?
If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of complex numbers?

You are indeed correct, this quotient ring simply means the set of polynomials over $\mathbb{Q}$ put into equivalence classes according to their remainders modulo $f(x)$. This is in general a ring but if $f(x)$ is irreducible then you get a field.
Another way to think of this is that you have decided to take the free variable $x$ and declare that it now satisfies $f(x) = 0$ (i.e. that $x$ is an abstract root of $f(x)$). Thinking in this way it is now intuitive why $\mathbb{R}[x]/(x^2+1)$ is isomorphic to $\mathbb{C}$...you have forced $x$ to behave like the complex number $i$ (or even $-i$, but this doesn't matter).
