Amending process, subject of Article 5 of US constitution, is cumbersome and requires super-majorities both for proposal (2/3rds of both houses of Congress) and ratification (3/4ths of states, 38 states). Congress determines method of state consideration, either by state legislatures (26 of 27 have been ratified this way) or by state conventions (as was done in case of constitution itself and 21st amendment repealing 18th amendment prohibiting liquor).
2/3rds of state legislatures can petition Congress to call a national convention to consider amendments but this has never been used (actual language is 'application' not petition and some constitutional scholars suggest that means Congress could reject the application). Scholars of US constitution tend to be conservative and guarded about it and 'protective' of it. They generally look with disfavor on the petition approach, suggesting that a national convention could open the entire document up for review and they apparently don't want that to happen (1787 constitution is a product of just such a convention, delegates were sent to Philadelphia to try to 'fix' the Articles of Convention but they proceeded from the get-go to write an entirely new document, one more favorable to folks of their class).
In any event, amending process is one of a number of anti-democratic features of US constitution (although it is not uncommon to read that 'the people have a right to change the Constitution' in intro US gov't texts), intended - as were such mechanisms as bicameral legislature, executive veto, staggered elections, judicial appointment with unlimited tenure - to thwart popular majorities. As James Madison (the 'father of the US Constitution) wrote, in *Federalist #10*, the document was intended to offer the appearance of democracy sans the substance. Michael Hoover