This question is probably more suitable for MSE. However, since it is still open and it has (currently) three upvotes, let me shortly answer it. Of course, my answer is just an extended version of the comments above, and everything I will say can be found in the classical literature on the subject, such as Beauville, Hartshorne, Barth-Hulek-Peters-Van de Ven, Griffiths-Harris, etc.
The intersection pairing $(C, D)$ on the Picard group $\mathrm{Pic}(X) \simeq H^1(X, \, \mathcal{O}_X^*)$ of a a smooth, complex projective surface $X$ is induced by the intersection pairing in $H^2(X, \, \mathbb{Z})$, in the sense that $$(C, \, D)=c_1(C) \cup c_1(D) \in H^4(X, \, \mathbb{Z}) \simeq \mathbb{Z},$$ where $c_1 \colon \operatorname{Pic}(X) \to H^2(X, \, \mathbb{Z})$ is the first Chern class map, seen as the connecting homomorphism arising from the exponential sequence $$0 \to \mathbb{Z} \to \mathcal{O}_X \to \mathcal{O}_X^* \to 0.$$
In particular, the intersection number $(C, \, D)$ is always a (possibly negative) integer and it cannot be infinite.
When $C$, $D$ are holomorphic curves with no component in common then, by the Poincaré duality $H^4(X, \, \mathbb{Z}) \simeq H_0(X, \, \mathbb{Z})$, we are really counting the intersection points of $C$ and $D$ (with multiplicity); in fact, the natural orientation given by the complex structure tells us that each intersection point $p \in C \cap D$ must be counted with positive sign.
It follows that, if $(C, \, D)$ is negative, then the two curves $C$, $D$ must have a component in common; this means in particular that the set $C \cap D$ is infinite (but $(C, \, D)$ is not!).
As a typical example, consider the exceptional divisor $E$ of a smooth blow-up: then $E^2=-1$, even if $E \cap E =E$ is an infinite set. In fact, in general, the self-intersection number $(C, \, C)$ can be interpreted as the degree of the normal bundle $N_{C/X} \simeq \mathcal{O}_C(C)$ of $C$ in $X$.
