First, two-thirds of Congress, both the House and the Senate must agree an amendment to the Constitution is necessary. A number of Amendments die in the Senate, as it is hard to get 67 of 100 Senators to agree on the need or usefulness of an Amendment.
If Congress determines an amendment is necessary there are two paths the amendment process can take. One path is to call a Constitutional Convention. This path has never been used, as a Constitutional Convention is not bound to address only the Amendment at issue, but can go further. The concern is that the entire Constitution could be rewritten completely.
The other path is to refer it to the States. Each State Legislature must consider the Amendment. Upon ratification by two-thirds of the States (38 of 50), the Amendment becomes effective. For many of the modern Amendments, States are given seven years to ratify the Amendment. This time limit is into required and more or less time can be provided.
There have been Twenty Seven Amendments to the Constitution. The first ten, the Bill of Rights, were adopted politically in conjunction with ratifying the Constitution to further limit the power of the Federal Government. Since 1795, when the eleventh Amendment was ratified, only 17 Amendments have been passed. This includes prohibition of liquor and the repeal of prohibition.
A Constitutional Amendment is always possible. However, it is intentionally very difficult to change the Federal Constitution and any Amendment is not likely without it being desired by a significant majority of the Country.