the classical matrix groups

The most concrete Lie groups are groups of matrices: collections of n-by-n matrices closed under multiplication and inversion, sitting inside the space of all matrices as smooth surfaces cut out by algebraic conditions. They are the working examples through which almost everyone first meets Lie theory, because here multiplication is literal matrix multiplication and the Lie algebra is a space of matrices you can write down.

The cast: GL(n, R) and GL(n, C), the general linear groups of all invertible real or complex matrices (dimension n^2 over R, 2n^2 for the complex one over R). SL(n), the special linear group, those with determinant 1 (dimension n^2 - 1). O(n), the orthogonal group of real matrices preserving the dot product, A^T A = I (dimension n(n-1)/2); its identity component SO(n) adds det = +1. U(n), the unitary group of complex matrices preserving the Hermitian inner product, A^* A = I (dimension n^2); SU(n) adds det = 1 (dimension n^2 - 1). Sp(n) (the compact symplectic group) preserves a quaternionic-Hermitian form and has dimension n(2n+1). Each is a closed subgroup of some GL, hence a Lie group by Cartan's closed-subgroup theorem, and each defining equation (det = 1, A^T A = I, ...) is what cuts the manifold out and fixes its dimension.

These groups, together with five exceptional groups, give essentially all the compact simple Lie groups, so they are the backbone of structure theory and representation theory; they are also the symmetry groups of physics (SU(3) x SU(2) x U(1) of the Standard Model, the Lorentz group). One honest warning about conventions: 'Sp(n)' is overloaded — some authors mean the compact group of quaternionic dimension n (rank n), others mean Sp(2n, R) or Sp(2n, C), the noncompact real or complex symplectic group of 2n-by-2n matrices preserving a skew form. Always check which Sp is meant.