It was best used by DPS champions that would benefit from high attack speed and/or crit chance, like Tryndamere. It was most often bought late game because the crit chance stat needed other late game crit items to be effective.
Elixirs are often bought towards the end of long games when builds are all completed and the extra gold continues to accumulate. They can also be bought mid game to give a temporary stat advantage during ganks and teamfights.
Elixirs are automatically applied when purchased if there are already 6 items in the inventory.
Elixir of Agility was not available in Dominion.
The Piece of Green Candy was a seasonal item that replaced the Elixir of Agility during the Harrowing celebration. It was the same item with a different name and a picture.
The Green Tea Kettle was another seasonal item that replaced the Elixir of Agility during the Lunar Revel celebration.
The Fury Fungus was yet another seasonal item that replaced the Elixir of Agility during the Great Hunt.
It was removed because it primarily benefited AD carries that were already in the lead, making hyper carries even stronger, and pigeon-holing end-game AD builds to focus on critical strikes.