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?

Yes that's what it means.  Take the isomorphism $\phi: \mathbb{R}[x]/(x^2+1)\rightarrow \mathbb{C}$ by $\phi: (a+bx) \mapsto (a+bi)$.  I'll leave it to you to show that it is an isomorphism.  
