How broad is the president's power of amnesty?

[zachs2]

All the Constitution says is that they "shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment." That seems very broad, but I want to clarify exactly how it works. If, for example, there were a president who decided that they wanted immigration reform or legal marijuana but Congress was stonewalling them, would they be able to just unilaterally declare an amnesty for all illegal immigrants or federal marijuana charges? If they did, would there be any possible legal recourse or is that amnesty just there to stay?