President Obama recently announced that he considers the Defense of Marriage Act (DOMA) to be an unconstitutional law, and he therefore will no longer allow the Department of Justice to defend the law in the courts. Since that decision, we have received a flurry of inquiries about whether the President has the constitutional authority to declare a law unconstitutional. The answer is, "Yes; he does have that power" – a point frequently reaffirmed by the Founders.
Granted, most Americans believe that the President's decision about the constitutionality of DOMA was wrong (and certainly, from an original intent standpoint it definitely was), but that is a completely different issue from whether he has the constitutional authority to declare the law unconstitutional.
However, history and the constitution make clear that the President does not have the final say on what is constitutional (nor does any other single branch, whether Judicial or Legislative). The other two branches still have recourses if they disagree with the President.
For example, Congress can (and I believe will) intervene to appoint legal counsel to defend the law in Court. And the upper courts can still find the law to be completely constitutional (as they have in previous challenges to the law), regardless of whether the Executive Branch chooses to enforce the law or defend it in court.