Exponential Map
Definition. $\exp: \mathfrak{g} \to G$; for matrix groups, the matrix exponential $e^X = \sum X^k/k!$. The curve $t \mapsto e^{tX}$ is the one-parameter subgroup with initial velocity $X$.
Working examples. $\mathfrak{u}(1) \to U(1)$: $i\theta \mapsto e^{i\theta}$ — the unit circle parametrized by angle. $\mathfrak{su}(2) \to SU(2)$: $e^{i\theta \hat{n}\cdot\vec\sigma/2}$ = rotation by $\theta$ about $\hat n$ (note the famous $\theta/2$).
Intuition. Rebuilds finite transformations from infinitesimal generators — "integrating" the algebra back to the group. Surjective onto the identity component for compact groups; a local diffeomorphism near $0$.
Physics anchor. Every "$e^{i\theta(x)}$" gauge factor in QM. Parallel transport is a path-ordered exponential $P\exp(-\oint \omega)$ of the connection — the exponential map applied along a curve. For $U(1)$, path-ordering is unnecessary and holonomy is just $e^{i\oint A}$.
Related: Lie Group · Lie Algebra · Parallel Transport and Holonomy · Module 03 - Lie Groups and Lie Algebras
Sources: Sternberg Ch. V; Woit Ch. on Lie groups/algebras.