I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:


What are some interesting examples of groups that aren't linear?


Are there general constructions that one can use to cook up a group or a family of groups that isn't linear?


Are there general techniques that one can use to show that a given group isn't linear?


More loosely, what is it about these groups that makes them interesting?



Understanding groups that are not linear
Consider the class of finitely generated linear groups. Such groups $G$ satisfy certain well-known restrictions, for instance:


1. 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.
2. $G$ 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.
3. $G$ 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](http://mate.dm.uba.ar/~gminian/atcba/talks/prassidis.pdf) 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 $Mod\_g$, $g\ge 3$. 


