That is way too ambitious, I think, considering what is known about classification of algebraic varieties. Ignoring the trivial, for these purposes, case of curves, the next and best studied case is algebraic surfaces. For them, the classification is classical and has been known for almost a century. So that is a good first case to consider.
So you are asking: what are the smooth projective surfaces $X$ with $\pi_1(X)=1$, and in particular with regularity $q=h^1(\mathcal O_X)=0$. Well, pick up your favorite text, Shafarevich etc. or Barth-(Hulek-)-Peters-van de Ven, or whatever, and go through the list.
Kodaira dimension $-\infty$: here you get all rational surfaces, and only these.
Kodaira dimension $0$: K3 surfaces.
Kodaira dimension $1$, elliptic surfaces $X\to C$ (a general fiber is elliptic). Clearly, $C$ must be $\mathbb P^1$. But actually getting $\pi_1(X)=1$ seems like not a completely trivial condition, something to think about.
Kodaira dimension $2$, i.e. surfaces of general type. Well, the examples of simply connected surfaces of general type are highly prized, especially if they have $p_g=h^0(\omega_X)=0$. Many such surfaces are known (e.g. http://en.wikipedia.org/wiki/Barlow_surface, some Godeaux surfaces, some Campedelli surfaces) but a complete classification? Not even close. Like I said, this is too ambitious for the present state of knowledge.