Consider the class of finitely generated linear groups. Such groups $G$ satisfy certain well-known restrictions, for instance:
Every such $G$ is residually finite (Malcev, 1940). Thus, most Baumslag-Solitar groups, e.g. $$ \langle a, b| a b^2 a^{-1} =b^3\rangle $$ are not linear. This is the simplest example of a nonlinear f.g. group I know.
$G$ is virtually torsion-free (Selberg, 1960). In particular, if $G$ is torsion then it is finite (which was known to Burnside). Note that there are infinite torsion residually finite groups (first examples are due to Golod and Shafarevich); such groups have to be nonlinear.
$G$ satisfies Tits' alternative (Tits, 1972): Either $G$ contains a free nonabelian subgroup or contains a solvable subgroup of finite index. (Thus, for instance, Thompson group is not linear.)
Tarski mosters will violate all of the above restrictions.
There are more subtle restrictions, for instance, $Aut(F_n), n\ge 3$ is not linear (Formanek and Procesi, 1992).
Consider reading Wehrfritz' book "Infinite linear groups" or this survey to get a better idea of what linearity means for f.g. groups, specially, Lubotzky's criterion of linearity.
Concerning your question of why nonlinear groups are interesting: Many of them occur naturally (like $Aut(F_n)$), the rest push the boundaries of our understanding of the class of f.g. groups. For many "natural" groups, linearity is unknown, e.g., the mapping class group $Mod_g$, $g\ge 3$.
