There are four ways that an amendment can be added to the Constitution. Specifically, there are two ways that an amendment can be proposed and two ways that a proposed amendment can be ratified. This means that there are four potential combinations.
With respect to the proposal of an amendment, an amendment can be proposed by Congress or by a special convention that is convened for the purpose of proposing amendments. If the amendment is proposed by Congress, two-thirds of each house must vote for it. A convention can be called by Congress if three quarters of the state legislatures request it. No convention has ever been called and therefore no amendment has ever been proposed in this way.
Once an amendment has been proposed, it must be approved by the states. There are two ways in which this can be done. First, the state legislatures can each vote on the amendment. Second, the states can each have a convention to vote on whether to ratify the amendment. Congress gets to decide which of these methods is used. The first method is by far the more common. In either case, three quarters of the states must approve the amendment in order for it to be added to the Constitution.