Over an algebraically closed field, smooth rational curves are trivial objects in themselves because they are isomorphic to ${\bf P}^1$. Once we consider morphisms of ${\bf P}^1$ to a given algebraic variety $X$, however, they involve interesting and subtle geometry of $X$. For example, we pose naive questions:
Q1. Given an algebraic variety $X$, determine whether $X$ contains rational curves of not.
Q2. If it does, how many rarational curves are there?
Partial answers are known. In case the canonical divisor $K_X$ of $X$ is not nef, $X$ contains at least one rational curve. If $K_X$ is negative, then there is a continuous family of rational curves which covers the whole $X$ (Mori, Kollár). The converse is of course not true. Indeed, Fermat hypersurfaces $X^n_d=\{x^d_0+x^d_1+\cdots x^d_{n+1}=0\}\subset {\bf P}^{n+1}$ of dimension $n$ and degree $d$ contains ${\bf P}^1$'s as linear lines. Every algebraic K3 surface $S$ is known to carry one or more rational curves (Mumford).
When $X$ is a Calabi-Yau 3-fold, counting the number of rational curves is a hot issue in conformal field theory. Experience and the posteulate of Boson-Fermion correspondence suggest that the number of rational curves of a given degree on $X$ (with respect to a polarization) is closely related to the moduli space of the "mirror variety" $X^*$ of $X$.
In general, it is very hard to answer the above two questions. Still there is a heuristic principle. If the canonical divisor $K_X$, contains "negative part", there are a lot of rational curves on $X$. On the contrary, if $K_X$ is "big", $X$ will not carry too many rational curves (a part of a conjecture of S. Lang, which asserts that all non-trivial holomorphic maps from ${\bf C}$ to $X$, a smooth variety of general type, will factor through maps to $Y\subset X$, a proper Zariski closed subset).
In this address, I will review some of the theory of rational curves on algebraic varieties and its application to the intrinsic algebraic geometry.
To make things simple, every varieties are assumed to be defined over
the complex numbers.
Theorem 1 (Cone theorem, due to S. Mori, Y.
Kawamata, J. Kollár, ...). Suppose that the canonical divisor
$K_X$ of a normal, projective, ${\bf Q}$-factorial variety $X$ with
only canonical singularities is not nef. Then the closure
$\overline{NE}(X)$ of the convex cone generated by effective 1-cycles
with real coefficients is locally polyhedral in the half space
$N^-_1(X)=\{C\in N_1(X)_{\bf R};(C,K_X)<0\}$ is and each extremal ray
$R$ in $N^-_1(X)$ of $\overline{NE}(X)$ corresponds to a projective
contraction morphism with nice properties. When $X$ is non-singular,
$R$ is generated by a rational curve $C$ such that $0<(C,-K_X)\leq\dim
X+1$.
As an immediate corollary, we see that a smooth projective variety with non-nef canonical divisor contains a rational curve.
If the minimal model program is presumed, every variety is birationally equivalent to a minimal model (with nef canonical divisor) or to a fibre space, with general fibres being ${\bf Q}$-Fano varieties with only terminal singularities. A minimal model is never covered by a family of rational curves, or, in other words, not uniruled. To the contrary, every ${\bf Q}$-Fano variety is uniruled (Kollár, Miyaoka-Mori).
It is thus a natural question to ask if the $n$-dimensional ${\bf
Q}$-Fano varieties with only terminal singularities are bounded.
Otherwise, the study of birational structure of uniruled varieties
would be extremely messy beyond control.
Theorem 2 (Campana, Kollár-Miyaoka-Mori). The
$n$-dimesional smooth Fano varieties form a bounded family. The ${\bf
Q}$-Fano 3-folds with only terminal singularities are also bounded.
A standard method to prove this kind of statement is to analyse rational curves of low degree. A similar technique applies to yield a general structure theorem of uniruled varieties, which asserts that a uniruled variety is canonically a tower of rationally connected varieties over a non-uniruled variety.
A variety is said to be rationally connected if its two general two
points can be joined by a rational curve on it. A smooth Fano variety
is rationally connected (Campana, Kollár-Miyaoka-Mori) but a ${\bf
Q}$-Fano variety (with non-terminal singularities) are not in genreal.
Theorem 3 (Kollár-Miyaoka-Mori). Let $X$ be a
uniruled variety. Then there exists a dominant rational map $\pi\colon
X\to Y$ such that a general fibre of $\pi$ is rationally connected and
that if $x_1,x_2\in X$ are mapped to two general and distinct points on
$Y$, then $x_1$ and $x_2$ cannot be joined by a rational curve on $X$.
This $\pi$ is uniquely determined by $X$ up to birational equivalence.
Fano varieties are viewed as natural genrealizations of projective spaces in the context of minimal model program. But the complete classification of such varieties is hopeless since the list would be tremendous in dimension four or more. We should rather be content with an abstract boundedness result and certain characterizations of specific varieties in terms of simple invariants.
For a smooth Fano manifold $X$, the minimum of the degrees (with
respect to the ample divisor $-K_X$) of the rational curves is an
invariant of $X$, called the length. The length of $X$ is known to be
at most $\dim X+1$ (Mori).
Theorem 4 (Cho-Miyaoka). Let $X$ be an
$n$-dimensional smooth Fano variety. Then the length of $X$ is $n+1$
[resp. $n$] if and only if $X$ is isomorphic to the projective
$n$-space [resp. $n$-dimensional hyperquadric].
These characterizations of projective spaces and hyperquadrics imply significant prior results like Hartshorne's conjecture and a theorem of Kobayashi-Ochiai.