All amendments to the Constitution have been proposed via the first method outlined in Article V of the Constitution, that is, proposed by 2/3 of both houses of Congress. The other method, which we have never used, is a national convention called by Congress at the request of 2/3 of the state legislatures.
We have never used that, but how close have we come? Were we ever just a few states away from a convention being called? Was there ever national media attention paid to the possibility that one would be called? Why haven't we ever done it? If there have been attempts, are there typical political leanings of the groups that pursue it? Or has it been typically bipartisan? Or varying?
There are three big problems with an Article V convention:
The difficulty of getting a supermajority.
The uncertainty posed by a lack of clear procedures for a convention.
The possibility of a "runaway convention."
Each of these issues causes a lot of uncertainty and makes this option relatively unattractive and difficult.
Supermajority
Simply, getting a 2/3 majority in both houses is hard enough, getting 2/3 of the States is harder.
Lack of Clear Procedures
The only Constitutional Convention we have as an example is the original one, and no procedures are spelled out in the Constitution. This means we wouldn't really know how to run a convention if we started one, and there would certainly be a lot of debate and consternation over the procedures that should be involved.
For example, what constitutes a valid application by a state legislature for a national convention? What procedures must a state follow in submitting an application? Must the precise language of the proposed amendment be included within the application? How similar must the language be in the applications of various states in order to permit Congress to count them? How long does an application by a state remain valid? May a state rescind its application? If so, under what conditions? What is the extent of Congress' power to review state applications? What institution of government controls the agenda of the convention-the state legislatures, Congress, the convention itself? May Congress refuse to submit the work product of the convention to the states for ratification and, if so, under what circumstances? How will delegates to the convention be selected? How will votes at the convention be counted? How will other procedures for the conduct of the convention be established? How will the convention be financed? If Congress assumes the power to answer some or all of these questions, are its determinations subject to review by any other institution of government, such as the courts? reference
Runaway Convention
This is the text of Article V:
The Congress, whenever two thirds of both houses shall deem it necessary, shall propose amendments to this Constitution, or, on the application of the legislatures of two thirds of the several states, shall call a convention for proposing amendments, which, in either case, shall be valid to all intents and purposes, as part of this Constitution, when ratified by the legislatures of three fourths of the several states, or by conventions in three fourths thereof, as the one or the other mode of ratification may be proposed by the Congress; provided that no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article; and that no state, without its consent, shall be deprived of its equal suffrage in the Senate.
Note that this does not limit the scope of a convention. It is sometimes claimed that a convention has to be called with a specific purpose in mind, and it must stick to that purpose. But there is no actual Constitutional requirement to do so. This means that a convention could potentially go far beyond its original mandate and make sweeping changes to the Constitution. Of course, the proposed amendment would still need to be ratified, but it does cut Congress out of the process entirely, and that makes a lot of people very uncomfortable.
There is one obvious historical example of a "runaway convention": the original Constitutional Convention. It was convened for the purposes of amending the Articles of Confederation to address various deficiencies in the Articles. It ended up producing an entirely new system of government, and proclaimed that ratifying this new Constitution required the consent of only three-fourths of the states, instead of the Article's required unanimous consent of state legislatures.
The legality was discussed in Federalist Number 40. Their argument is basically:
The convention was gathered to fix the problems of the Articles, and they deemed this was the only way to fix them, therefore it was proper.
Ignoring the Article's required unanimous consent was okay because having the people's permission is more important than the Articles' requirement of the consent of the state legislatures.
So it's not impossible a convention could be called to discuss, say, term limits, and end up with radical changes because the convention deems that the underlying reason that term limits were requested stems from other deficiencies in the Constitution.
There's been tons of discussion about these issues. See the above referenced article, and this Congressional Research Service article.