A divisor $D$ is an element of the quotient sheaf $\mathcal{R^*}/\mathcal{O^*}$ on a compact complex manifold $X$. They can be also thought of via valuations as formal sums of hypersurfaces.
A complete linear system $\mathcal{L}(D)$ is defined to be all positive divisors equivalent to $D$. These are bijective with the subspace of meromorphic functions $f$ such that $(f)+D\geq0$. This is bijective with $\mathbb{P}(H^0(X,[D]))$ (Holomorphic sections of the line bundle corresponding to D). A linear system is a subset of $\mathcal{L}(D)$ corresponding to a subspace of either of the last two sets.
The base locus of such a system of positive divisors is the set of divisors which occur in all the divisors in the system.
Bertini's theorem states that the generic element of a linear system is smooth away from the base locus of the system. This means that on a dense open set in the vector spaces above, the corresponding divisors are non-singular away from the base locus.
How do I use this to prove the more familiar fact that given a submanifold $M$ of some $\mathbb{P}^N$, there is a hyperplane $H=\mathbb{P}^{N-1}$ with $H\cap M$ non singular.
I hope someone can make some suggestions on how to proceed.
This is just paraphrasing Hartshorne's proof. Look at the hyperplanes tangential at a point p∈M. This is a codimension N−dimM−1 linear subspace in the dual space $\mathbb{P}^{N∗}$. So, as p varies, it sweeps out a subvariety of the dual space of dimension at most N−1. So, general hyperplane is transversal at every point of M.