A few of the more obvious ones:

* Resolution of singularities in characteristic p
*Hodge conjecture
* Standard conjectures on algebraic cycles (though these are not so urgent since Deligne proved the Weil conjectures).
*Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details.

For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces.

A few of the more obvious ones:

* Resolution of singularities in characteristic p
*Hodge conjecture
* Standard conjectures on algebraic cycles (though these are not so urgent since Deligne proved the Weil conjectures).
*Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details.

For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces.

(Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. It is classical that the moduli space is unirational for genus at most 10, and I think this has more recently been pushed to genus about 13. Mumford and Harris showed that it is of general type for genus at least 24. As far as I know most of the remaining cases are still open.