a Lie group
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Think of all the ways you can rotate a globe. There are infinitely many rotations, and they vary smoothly: a tiny twist of the wrist produces a rotation close to the one before, and you can slide continuously from any rotation to any other. At the same time, rotations compose like a group — do one, then another, and you get a third, with an identity (do nothing) and an inverse (undo). A Lie group is exactly this marriage: a set that is simultaneously a smooth manifold and a group, where the two structures cooperate.
Precisely, a Lie group G is a smooth manifold equipped with a group law such that multiplication m: G x G -> G, (a, b) |-> a*b, and inversion i: G -> G, a |-> a^(-1), are both smooth maps. (Smoothness of multiplication alone forces inversion to be smooth too.) The dimension of G is its dimension as a manifold. The circle S^1, with angle addition, is a 1-dimensional Lie group. The rotation group SO(3) is 3-dimensional. The real line R under addition, and the nonzero reals R^* under multiplication, are abelian Lie groups. Crucially, every point looks like every other: left multiplication by a fixed g, the map L_g(x) = g*x, is a diffeomorphism carrying the identity e to g, so the local geometry near e is copied everywhere. That is why a Lie group's entire infinitesimal structure is captured at one point — the identity.
Lie groups are the precise language of continuous symmetry, which is why they pervade geometry and physics: the Lorentz group, gauge groups, the symmetries of a homogeneous space, the structure group of a bundle. A common subtlety: a Lie group need not be connected (the orthogonal group O(n) has two components, det = +-1) and need not be compact (GL(n,R) is open in matrix space, hence noncompact). 'Lie group' carries no topological promises beyond being a smooth manifold and a group with smooth operations.
The unit circle S^1 = {e^(i*theta)} with complex multiplication: multiplying e^(i*alpha) by e^(i*beta) gives e^(i*(alpha+beta)), a smooth operation; the identity is 1 and the inverse of e^(i*theta) is e^(-i*theta). It is a compact, connected, 1-dimensional abelian Lie group, often written U(1).
S^1 = U(1): the simplest nontrivial Lie group, and the prototype of a one-parameter group.
Do not assume connectedness or compactness from the word 'Lie group'. O(n), GL(n), and the Lorentz group are all genuine Lie groups that are disconnected or noncompact (or both).