The United States Constitution may be amended either by a two-thirds vote by the House of Representatives and the Senate followed by subsequent state level ratification or by a Convention called for by two-thirds of state legislatures and subsequent state level ratification. To date, only the former method has been implemented.
In both instances, the support of three quarters of the state legislatures (or 38 states as of 2014) is required to ratify a proposition.
Amendments to the United States Constitution are therefore notoriously difficult to make, since only 13 states have to oppose an amendment for it to be thrown out. For this reason, only 27 amendments have ever been made, 10 of which made up the Bill of Rights. This enshrines the rights of free speech, press, religion, bearing arms and fair trial by jury, among others.
The implication of this difficulty is that the Supreme Court has a great responsibility in its interpretation of the Constitution in order to make decisions, since it is unlikely that their decisions will ever be overturned by an amendment. An example of an unpopular Supreme Court decision that nevertheless stood due to a voted-down amendment was their defense of flag burning. In 2006, the Senate's proposed amendment to make flag burning illegal was voted down by 66 state legislatures to 34.
The specific article of the Constitution pertaining to making amendments is Article V.