SU(2) and SO(3)
Definition. $SU(2)$ = unitary $2\times2$ matrices of determinant 1; as a manifold, $S^3$. $SO(3)$ = rotations of $\mathbb{R}^3$. The Adjoint Representation gives a 2-to-1 surjection $SU(2) \to SO(3)$ with kernel $\{\pm I\}$: the double cover.
Intuition. $SU(2)$ is "rotations that remember whether you've turned $2\pi$ or $4\pi$" — the belt trick. Same Lie Algebra ($\mathfrak{su}(2) \cong \mathfrak{so}(3)$), different global topology — the canonical warning that the algebra does not determine the group.
Physics anchor. Spin-$\tfrac12$: fermions carry representations of $SU(2)$, not $SO(3)$ ($\psi \to -\psi$ under a $2\pi$ rotation). Woit develops this at length.
Role in this study. Your first nonabelian group: the practice case for Yang-Mills Theory, where $[A, A] \neq 0$ makes the field self-interacting. Also, as a manifold $SU(2) = S^3$ is the total space of the Hopf Fibration — the same sphere serves double duty in the monopole story.
Related: Lie Group · Adjoint Representation · Group Representation · Hopf Fibration · Yang-Mills Theory
Sources: Woit (SU(2) chapters); Sternberg Ch. V.