Lets start with the basics. Suppose that $X(t)$ is a time-homogeneous Markov process. We define the transition semigroup of this process by $S(t)f(x) = \mathbb{E}[f(X(t)) \,|\, X(0) = x]$. (i.e. we have $S(0)$ is the identity and $S(t) S(r) = S(t+r)$ by the Markov property).
We can calculate the infinitesimal generator of this semigroup, which is the derivative of the semigroup at $t=0$: $Lf = \frac{d}{dt}|_{t = 0} S(t)f = \lim_{t \rightarrow 0} \frac{S(t)f - f}{t}$ defined for $f \in \mathcal{D}(L)$, the set of functions for which this limit exists. See http://en.wikipedia.org/wiki/C0-semigroup.
Skipping technicalities, we get by the vector valued version of the fundamental theorem of calculus that for $f \in \mathcal{D}(L)$
$$
S(t)f - f - \int_0^t L S(r)f d r = 0,
$$
which implies that for $f \in \mathcal{D}(L)$
$$
f(X(t)) - f(X(0)) - \int_0^t L f(X(s)) d s
$$
is a martingale.
$L$ is an important object because $L$ tells you what the process is doing. For example we have a process $X(t)$ on $\mathbb{R}$ such that
$$
L^Xf(x) = f''(x) + b f'(x)
$$
then the process is a diffusion process with drift $b$. A process $Y(t)$ on $\mathbb{N}$ such that its generator satisfies
$$
L^Y f(n) = f(n+1) - f(n)
$$
is a Poisson jump process.
In other words, every(reasonably behaved) time-homogeneous Markov process gives you a generator $L$.
For applications, one might want go the other way around. Suppose you want to model a phenomenon that you expect to behave Markovian. A first step is to write down an operator $L$ that should serve as the generator of the Markov process.
However, a priori it is not clear that every operator $L$ corresponds to a Markov process. This is the Martingale problem: given an operator $(L, \mathcal{D}(L))$, find a process $X(t)$ such that for $f \in \mathcal{D}(L)$
$$
f(X(t)) - f(X(0)) - \int_0^t L f(X(s)) d s
$$
is a martingale. If you succeed in doing this, then you have a process that behaves as your operator $(L,\mathcal{D}(L))$ describes.
A strategy to solve the martingale problem is by taking approximating generators $L_n$, i.e. $L_n \rightarrow L$ for some topology on the space of operators, for which it is known that there are Markov process $X^n(t)$ solving the martingale problem for $L_n$(for example the $L_n$ are bounded). Associated to these Markov processes $X^n(t)$ there are probability measures on the trajectory space, lets denote them with $\mathbb{P}^n$. Now one has to do two things.
First, one shows that the sequence $\mathbb{P}^n$ has limit points[existence].
Second, one shows that there is at most one limit point[uniqueness].
The fact that $L_n \rightarrow L$ and some technicalities then implies that this unique limit point is a solution to the martingale problem for $L$.
Ps. One can also try to show that $(L,\mathcal{D}(L))$ generates a semigroup via the Hille-Yosida theorem, and then construct a Markov process from the semigroup. This is a different and approach, but gives essentially the same result. Depending on the operator one approach is easier than the other.
