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 $m<n$. Then $$\dim CH^m(X)\otimes \mathbb{Q}>1$$
Proof: The class of the general fibre and $H^m$, with $H$ an ample divisor, are independent.
Cor: $\mathbb{P}^n$ does not surject onto a 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 note 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.