Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem.
They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations.
A finite group G of order n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ring K[G] by left (or right) multiplication is linear and faithful.
While example 4 above is too general to define a distinctive class (it includes all linear groups), restricting to a finite index set I, that is, to finitely generated groups allows to construct many interesting examples.
For example: In some cases the fundamental group of a manifold can be shown to be linear by using representations coming from a geometric structure.
Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above.