What are some open problems in algebraic geometry?
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.
Let me mention a couple of problems related to vector bundles on projective spaces.
The Hartshorne conjecture. In its weak form it says that any rank 2 vector bundle on $\mathbf{P}^n_{\mathbf{C}},n>6$ is a direct sum of line bundles, which implies that any codimension 2 smooth subvariety whose canonical class is a multiple of the hyperplane sectionis a complete intersection. In a stronger form Hartshorne's conjecture says that any codimension $>\frac{2}{3}n$ subvariety of $\mathbf{P}^n_{k},k$ an algebraically closed field is a complete intersection. See Hartshorne, Varieties of small codimension in a projective space, Bull AMS 80, 1974. The weak conjecture fails for $n=3$ and $4$ -- there are examples (due to Horrocks and Mumford) of non-split vector bundles of rank 2 on $\mathbf{P}^4_{\mathbf{C}}$, but so far as I know the question if any such examples exist for $n>4$ is open. See here Evidences on Hartshorne's conjecture? References? for a discussion including some references.
The existence of non-algebraic topological vector bundles on $\mathbf{P}^n_{\mathbf{C}}$. It is a classical result that any topological complex vector bundle on $\mathbf{P}^n_{\mathbf{C}}, n\leq 3$ is algebraic, see e.g. Okonek, Schneider, Spindler, Vector bundles on complex projective spaces, chapter 1, \S 6. It is strongly suspected that for $n>3$ there are topological complex vector bundles that are not algebraic. Good candidates are nontrivial rank 2 vector bundles on $\mathbf{P}^n_{\mathbf{C}}, n\geq 5$ all of whose Chern classes vanish which were constructed by E. Rees, see MR0517518. It is claimed there that these bundles do not admit a holomorphic structure, but later a gap was found in the proof. See here Complex vector bundles that are not holomorphic for some more information.
The Jacobian conjecture