Lie Group
Definition. A group $G$ that is also a Smooth Manifold, with multiplication $(g,h) \mapsto gh$ and inversion $g \mapsto g^{-1}$ smooth.
Working examples. $U(1) = \{e^{i\theta}\}$ (the circle); $SU(2)$ (the 3-sphere); $SO(3)$; $GL(n,\mathbb{R})$. For this study, matrix groups are all you need.
Intuition. Continuous symmetries. The manifold structure lets you differentiate along the group — giving the Lie Algebra — and the group structure lets you translate: everything about $G$ near any point is determined by what happens near the identity.
Role in bundle theory. $G$ appears as the structure group: Transition Functions of a bundle take values in $G$, and a Principal Bundle has $G$ itself as fiber. Choosing $G$ = choosing which force: $U(1)$ → electromagnetism, $SU(2)\times U(1)$ → electroweak, $SU(3)$ → strong.
Related: Lie Algebra · Exponential Map · U(1) · SU(2) and SO(3) · Group Representation · Module 03 - Lie Groups and Lie Algebras
Sources: Sternberg Ch. V; Woit (early chapters).