As Damiano pointed out, if a smooth projective $X$ surjects onto a curve, and $\dim X>1$ then the Picard number is at least $2$. This gives a clear obstruction, which explains why $\mathbb{P}^n$
won't satisfy it (when $n>1$). Here is a small refinement:

Lemma: Suppose that an $n$ dimensional variety $X$ surjects onto a $m$-dimensional variety with $0<m<n$. Then 
$$\dim CH^m(X)\otimes \mathbb{Q}>1$$

Proof: The class of the general fibre and $H^{n-m}$, with $H$ an ample divisor, are independent.

Cor: $\mathbb{P}^n$ does not surject onto a nontrivial lower dimensional variety.

(This can also be seen directly from Bezout's theorem.)

 On the other hand,
$X$ will map onto $\mathbb{P}^1$ after blowing up, as Charles observed, and this is often
a very useful trick in practice.

The case of (`*`) $X$ mapping  onto curves of genus two or more is actually something that has been studied a number of people*. 
From Castelnuovo-De Francis (see Damiano's answer) one can extract a number of topological criteria. Here's one: $X$ satisfies (`*`) if and only if the fundamental group admits a surjective homomorphism onto the fundamental group of such a curve. (In my original answer,
I hadn't realized that JVP already discussed this.)
Some of this probably described in the multi-author book on Kaehler groups.
Also take a look at my  <a href="http://arxiv.org/abs/math/9204225"> note </a> in the
Bulletin from way back in the last century. 

In case it wasn't clear the results
in the last paragraph are characteristic $0$ only. The positive characteristic case has not
been looked at seriously, as far as I know, and is potentially very interesting. Warning: the
standard tecnhiques such as Castelnuouvo-De Franchis, will fail!


`*` Among them: Amoros, Beauville, Bressler, Campana, Catanese, Gromov, Green,
Lazarsfeld, Ramachandran, Simpson Siu, and me. People coming from hyperplane
arrangement theory have also looked at the question
for open varieties, but here the literature is so large, I won't even attempt it.

<hr>
The last edit was to rule out maps to points. I'm sure there is still some stupid
exception that I'm overlooking.