It is best used by casters or other champions that scale with ability power, but the cooldown reduction is useful on every champion.
This is a great item for manaless champions that lack the Crest of the Ancient Golem and need cooldown reduction.
Using Elixir of Brilliance during the mid and late game is a good way to reach the 40% cooldown reduction cap without having to buy extra items.
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, or even at level one.
The Piece of Blue Candy was a seasonal item that replaced the Elixir of Brilliance during the Harrowing celebration. It was the same item with a different name and picture.
The Blue Lasi Kettle was another seasonal item that replaced the Elixir of Brilliance during the Lunar Revel celebration.
The Sagely Spores was yet another seasonal item that replaced the Elixir of Brilliance during the Great Hunt.