Going all the way back to Fork, there have been many many ways to put a copy of another spell on the stack. Most often this is pretty simple, you just put another copy of the original target on the stack. You are generally allowed to change the targets for this copy but the copy is treated as the same as the original. (With Fork, this copy is red regardless of the colors of the original.)

One very important thing to note is that copies of spells that are put on the stack are not cast, they are just put on the stack. Spells and abilities that trigger or care about spells being cast will not trigger or count this copy as being played, as it is not played. For example, if a copy a spell is created by Echo Mage‘s ability, Painful Quandary will not trigger for the copy that is put on the stack. (It will trigger for the original spell being played, just not the copy.)

This is different from something like Isochron Scepter‘s ability where a copy of a card is created and this copy is cast. Because this spell is cast, abilities that trigger when a spell is cast will trigger. It does not matter that this spell is not represented by an actual card, it is cast.