Hopf fibration is a connection between state vector representation of a qubit $\lvert\psi\rangle\in\mathbb{C}^2$ and points on the Bloch sphere $(x, y, z)\in\mathbb{R}^3$.

Most compactly, the Hopf map $\pi$ can be written in the following way using Dirac notation, where $\sigma_i$ are Pauli matrices.

$$\pi:\mathbb{C}^2\rightarrow\mathbb{R}^3\qquad \pi(\lvert\psi\rangle) = \langle\psi\lvert\sigma_i\lvert\psi\rangle $$

Explicit coordinates on the Bloch sphere: $$ x = 2\mathfrak{Re}(a^*b)$$ $$ y = 2\mathfrak{Im}(a^*b)$$ $$ z = |a|^2 - |b|^2$$

A qubit is – in a nutshell – a construct of two complex numbers. We write it as a column vector in Hilbert space $\mathbb{C}^2$. But a qubit is commonly visualized as a point on the Bloch sphere. The Bloch sphere is a 3-dimensional sphere and the coordinates on its surface lie in 3-dimensional space $\mathbb{R}^3$. So how is it possible that a qubit can simultaneously be an object in $\mathbb{C}^2$ and $\mathbb{R}^3$? What connects these two? That connection is the Hopf fibration.