This question is actually a little bit vague. Anyway, I hope you can find the following answer useful.

One of the more general results about linear systems of curves on surfaces is the following theorem, proven by I. Reider using rank $2$ vector bundles techniques:

Theorem (Reider). Let $X$ be an algebraic surface, and $D$ be a nef and big divisor on $X$. Then

1. If $D^2 \geq 5$ and $x$ is a base point of $|K_X+D|$, then there exists a curve $E$ on $X$ with $x \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $DE=1$ and $E^2=0$.

2. If $D^2 \geq 10$ and $x,y$ are two points, possibly infinitely near, such that $|K_X + D|$ does not separate $x$ and $y$, then there exists a curve $E$ on $X$ such that $x,y \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $-2$ or $DE=1$ and $E^2=0$ or $1$ or $DE=2$ and $E^2=0$.

This result has many important consequences. For instance, it can be used to deduce Bombieri's theorem for pluricanonical systems (if $X$ is a surface of general type, then the $5$-canonical map is a birational morphism of $X$ onto its canonical model $X^{\textrm{can}}$.

Another application is the following result for abelian surfaces:

Theorem. Suppose $D$ is an ample line bundle of type $(1,d)$, with $d \geq 5$, on an abelian surface $X$. Then the morphism $$\varphi_D \colon X \to \mathbb{P}^{d-1}$$ is an embedding if and only if there is no elliptic curve $E$ on $X$ with $ED=2$.

This question is actually a little bit vague. Anyway, I hope you can find the following answer useful.

One of the more general results about linear systems of curves on surfaces is the following theorem, proven by I. Reider in his paper Vector bundles of rank $2$ and linear systems on algebraic surfaces (Annals of Mathematics 127):

Theorem (Reider). Let $X$ be an algebraic surface, and $D$ be a nef and big divisor on $X$. Then

1. If $D^2 \geq 5$ and $x$ is a base point of $|K_X+D|$, then there exists a curve $E$ on $X$ with $x \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $DE=1$ and $E^2=0$.

2. If $D^2 \geq 10$ and $x,y$ are two points, possibly infinitely near, such that $|K_X + D|$ does not separate $x$ and $y$, then there exists a curve $E$ on $X$ with $x,y \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $-2$ or $DE=1$ and $E^2=0$ or $1$ or $DE=2$ and $E^2=0$.

This result has many important consequences. For instance, it can be used to deduce Bombieri's theorem for pluricanonical systems (if $X$ is a surface of general type, then the $5$-canonical map is a birational morphism of $X$ onto its canonical model $X^{\textrm{can}}$.

Another application is the following result for abelian surfaces, see [Birkenhake-Lange, Complex Abelian Varieties, Chapter 10]:

Theorem. Suppose $D$ is an ample line bundle of type $(1,d)$, with $d \geq 5$, on an abelian surface $X$. Then the morphism $$\varphi_D \colon X \longrightarrow \mathbb{P}^{d-1}$$ is an embedding if and only if there is no elliptic curve $E$ on $X$ with $ED=2$.

When $D$ is very ample, one can also use the following classical result, known as "adjunction theorem" and whose modern form is due to Sommese [Hyperplane sections of projective surfaces I - The adjunction mapping. Duke Math. J. 46 (1979)]:

Adjunction Theorem. Let $X \subset \mathbb{P}^n$ be a smooth surface and $D$ the hyperplane class. Then $|K_X+D|$ is not special and has dimension $N=g(D)+p_g(X)-q(X)-1$. Moreover

$(A)$ $|K_X+D|= \emptyset$ if and only if

$(A1)$ $X \subset \mathbb{P}^n$ is a scroll over a curve of genus $g=g(D)$ or

$(A2)$ $X= \mathbb{P}^2$, $D=\mathcal{O}_{P^2}(1)$ or $D=\mathcal{O}_{P^2}(2)$.

$(B)$ If $|K_X+D| \neq \emptyset$ then $|K_X+D|$ is basepoint free. In this case $(K_X+D)^2=0$ if and only if

$(B1)$ $X$ is a Del Pezzo surface $($in particular $X$ is rational$\,)$ or

$(B2)$ $X \subset \mathbb{P}^n$ is a conic bundle.

If $(K_X+D)^2>0$ then the map $$\varphi_{|K_X+D|} \colon X \longrightarrow X' \subset \mathbb{P}^N$$ defined by $|K_X+D|$ is birational onto a smooth surface $X'$ of degree $(K_X+D)^2$ and blows down all the lines $E$ on $X$ such that $K_XE=-1$, unless

$(1)$ $X=\mathbb{P}^2(p_1, \ldots, p_7), \quad D=6L-\sum_{i=1}^7 2E_i$,

$(2)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=6L-\sum_{i=1}^7 2E_i -E_8$,

$(3)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=9L-\sum_{i=1}^8 3E_i$,

$(4)$ $X= \mathbb{P}(\mathcal{E})$, where $\mathcal{E}$ is an indecomposable rank $2$ vector bundle over an elliptic curve and $D=3B$, where $B$ is an effective divisor on $X$ with $B^2=1$.