Hopf Fibration
Definition. $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$. Write $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\}$; the U(1) action $(z_1, z_2) \mapsto (\lambda z_1, \lambda z_2)$, $|\lambda|=1$, is free, and the quotient is $\mathbb{CP}^1 = S^2$. So $S^3$ is a Principal Bundle over $S^2$ with group $U(1)$.
Why it's nontrivial. $S^3 \neq S^2 \times S^1$ (e.g. $\pi_1(S^3) = 0$ but $\pi_1(S^2 \times S^1) = \mathbb{Z}$). Hence no global section exists — the canonical example of a principal bundle with no possible global gauge.
The picture. $S^3$ decomposes into linked circles, one over each point of $S^2$; any two Hopf circles are linked once. (Worth watching a visualization once — search "Hopf fibration animation".)
Physics: this IS the Dirac monopole. The Hopf bundle is the $c_1 = 1$ $U(1)$-bundle over $S^2$; a sphere around a unit Dirac Monopole carries exactly this bundle. Charge-$n$ monopoles use its $n$-th powers. The natural connection on it has curvature = the uniform "magnetic field" on $S^2$ with total flux $2\pi$.
Bonus tie-in. $S^3 = SU(2)$ (SU(2) and SO(3)), and $\pi$ is the restriction of $SU(2) \to SU(2)/U(1)$.
Related: Principal Bundle · U(1) · Dirac Monopole · Characteristic Classes · Module 11 - Topology - Monopoles and Chern Classes