There are infinite groups which cannot be represented by finite-dimensional matrices over any commutative ring. If a group can be represented by matrices in such a way then it is called linear$^{\ast}$. Not all groups are linear.
A group $G$ is residually finite if for every element $g\in G$ there exists a homomorphism $\phi_g: G\rightarrow F_g$ onto a finite group $F_g$ such that $\phi_g(g)$ is non-trivial. Equivalently, for every element $g\in G$ there exists an action of $G$ on a finite object such that the action of $g$ is non-trivial. It is a rather famous result of Malc'ev that finitely generated linear groups are residually finite. This allows you to conjure up non-linear groups almost at will!
Non-Hopfian groups One way of constructing finitely-generated, non-residually finite groups is to construct finitely generated groups which have a surjective endomorphism $G\rightarrow G$ which is not an isomorphism. Such groups are called non-Hopfian, and it is a result of (again) Malc'ev that these groups are non-residually finite. See this Math.SE answer of mine for the proof, and this one for examples of such groups (the main example is the group $\langle a, b; b^{-1}a^2b=a^3\rangle$).
Simple groups A second way of constructing finitely-generated, non-residually finite groups is to construct finitely generated infinite simple groups. Examples of such groups are Thompson's group's $T$ and $V$ (these can be realised as groups acting on the unit interval in very natural ways - see these notes or this answer of mine) and Tarski monster groups.
Higman's group The group $G=\langle a, b, c, d; a^{-1}ba=b^2, b^{-1}cb=c^2, c^{-1}dc=d^2, d^{-1}ad=a^2\rangle$ was the first example of a finitely generated, infinite group with no finite quotients. This clearly implies that $G$ is not residually finite. Higman's paper is a joy to read$^{\dagger}$, and in it he points out that $G$ can easilly be used to construct finitely generated, infinite simple groups (his paper was pre-Thompson's groups, and pre-Tarski monsters) - taking a maximal normal subgroup $N$ of $G$, $G/N$ must be simple!
$^{\ast}$Perhaps this definition really requires field not commutative ring, but everything in this answer works for the more general commutative ring definition.
$^{\dagger}$Higman, Graham (1951), A finitely generated infinite simple group, J.Lon. Math. Soc.