In his thesis on K3 surfaces Saint-Donat proves the following fact (thm 6.1)
Let $L$ be a line bundle on a K3 surface $X$ such that the linear system $|L|$ has no fixed components and the morphism $\varphi_L$ is birational. Then the natural map $$S^\ast H^0(X,L)\rightarrow\oplus_{n\geq0} H^0(X,L^{\otimes n}) $$ is surjective.
Denote by $I$ the kernel of this map. Later in the paper (thm 7.2) Saint-Donat shows that $I$ is generated by elements of degree $2$ and $3$.
Now, my problem is that I do not understand the notation $S^\ast H^0(X,L)$, and as a consequence what is the map and why $I$ is the homogeneous ideal of the image surface $S=\varphi_L(X)$.
Any elaboration on these concepts would be very useful :)
As Hoot points out $S^\ast H^0(X,L)$ is the symmetric algebra of the finite-dimensional vector space $H^0(X,L)$. Elements of $S^\ast H^0(X,L)$ are polynomials with variables $s_0,...,s_n$ a base of sections from $H^0(X,L)$. The graded component of $S^\ast H^0(X,L)$ of degree $d \in N$ are all homogeneous polynomials of degree d. Furthermore, the canonical map
$$f: S^\ast H^0(X,L)\rightarrow\oplus_{n\geq0} H^0(X,L^{\otimes n})$$
is defined by taking the product of sections. Its image are those sections in $L^{\otimes n}$ which are - globally - the product of sections in $L$. If $L$ has no base-points, then its complete linear system defines the holomorphic map
$$\phi_L: X \longrightarrow \check{\mathbb P}(H^0(X, L)), x \mapsto [s \mapsto s(x)].$$
Here $\phi_L(x)$ is considered a linear functional on $H^0(X, L)$. The image $\phi_L(X) \subset \check {\mathbb P} (H^0(X, L))$ is the projective variety $Var(I)$ with
$$I := ker \ f \subset S^\ast H^0(X,L)$$
the ideal generated by all homogeneous polynomials which vanish on all points $x \in X$.
Probably it helps imagining $H^0(X,L)$ as $\mathbb C^{n+1}$. Then it could be easier to connect to the classical context of projective space and its subvarieties defined by homogeneous polynomials.
Note. There is a companion to Saint-Donat's paper by Francois C. Cossec: Projective Models of Enriques Surfaces. Math. Ann. 265, 283-334 (1983)