What are the ways amendments become laws?

"[There are] two methods of proposing amendments: by Congress or by a national convention requested by the states. Under the first method, Congress can propose an amendment by a two-thirds vote (of a quorum, not necessarily of the entire body) of the Senate and of the House of Representatives. Under the second method, two-thirds (2/3) of the state legislatures may convene and "apply" to Congress to hold a national convention, whereupon Congress must call such a convention for the purpose of considering amendments. As of mid-2006, only the first method (proposal by Congress) has been used. 

Once proposed—whether submitted by Congress or by a national convention—amendments must then be ratified by three-fourths (3/4) of the states to take effect. Article Five gives Congress the option of requiring ratification by state legislatures or by special conventions assembled in the states. The convention method of ratification has been used only once (to approve the 21st Amendment). Article Five currently places only one limitation on the amending power—that no amendment can deprive a state of its equal representation in the Senate without that state's consent."