Understanding groups that are not linear
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 $Mod_g$, $g\ge 3$.
The following is an elaboration of the last paragraph of Misha's answer.
For me, the thing that makes non-linear (discrete) groups interesting is that we are not very good at constructing them! Linearity remains unknown for many natural examples of groups, such as mapping class groups, and if true would dramatically simplify some hard theorems. (For instance, Daniel Groves has a very long and difficult proof that mapping class groups are 'equationally Noetherian'; if they are linear then it is an easy consequence of Hilbert's Basis Theorem.)
Similarly, we only know one way of constructing non-linear word-hyperbolic groups (noticed by Misha, in fact): take a uniform lattice $\Gamma$ in Sp(n,1), which is both word-hyperbolic and satisfies Margulis super-rigidity, and kill a 'random' element; the resulting quotient $Q$ is an infinite quotient of $\Gamma$ with infinite kernel, so by Margulis super-rigidity cannot be linear (at least in zero characteristic; I'm not sure about characteristic $p$).
The fact that we know no other methods of constructing non-linear groups is related to the fact that we do not know how to construct a non-residually finite word-hyperbolic group.
Surprising recent developments partially explain this failure by showing that linear (and hence residually finite) groups are much more common than we thought. I think most experts would have guessed that a 'random' finitely presented group (which is known to be word-hyperbolic) would not be linear; in fact, it follows from recent work of Agol and older work of Wise that some parts of the 'spectrum' of random groups are in fact linear. These are exciting times!
The universal covering $G^*$ of the group $SL_2({\mathbb R})$ is not linear. The reason is that any linear representation of $G^*$ is given by a Lie algebra representation, which descends to a representation of $SL_2({\mathbb R})$. The essential reason is that $SL_2({\mathbb C})$ is simply connected.
More generally, if $G$ is a simple algebraic group defined over ${\mathbb R}$ such that $G({\mathbb C})$ is simply connected, and if $G({\mathbb R})$ is not simply connected, then the universal cover of $G({\mathbb R})$ is not linear.