The "conjecture" in algebraic geometry that all rationally connected varieties are unirational comes to mind. It's usually thrown around as a way of saying, "See, we know so little about what varieties can be unirational that we can't prove a single rationally connected variety isn't." Unirationality implies rational connectedness, but I think almost everyone believes the converse should be false.

Some background: Algebraic geometers have been interested for a long time in proving that certain varieties are or are not rational (very roughly, figuring out which systems of polynomial equations can have their solutions parametrized.) Clemens and Griffiths showed in 1972 that a cubic hypersurface in $\mathbb{P}^4$ is irrational. Since then, there's been a lot of progress in rationality obstructions e.g., Artin-Mumford's obstruction via torsion in $H^3$, Iskovskikh-Manin on quartic threefolds, Kollar's work on rationality of hypersurfaces, and most recently, Voisin's new decomposition of the diagonal invariants which have led to major breakthroughs. 

On the other hand, unirationality has proved a far harder notion to control, and to my mind the biggest open question in this area is to find any obstruction to unirationality.