It is often used towards the end of the game to help with a final push or to help defend. It is best for a champion that relies heavily on physical damage, but it has benefit for all champions due to the additional health.
The health bonus can be used somewhat like Heal, instantly granting you extra life that could save you from death. This is a common tactic to turn the tide in an early fight or to survive a gank. For example, having one ready to use is an effective way to prevent a death early or mid game to Karthus'sRequiem.
Because the extra health is not gained by health regeneration or healing, it can be considered an effective counter to Ignite.
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.
The Piece of Red Candy was a seasonal item that replaced the Elixir of Fortitude during the Harrowing celebration. It was the same item with a different name and picture.
The Red Ginseng Kettle was another seasonal item that replaced the Elixir of Fortitude during the Lunar Revel celebration.
The Mightshroom was yet another seasonal item that replaced the Elixir of Fortitude during the Great Hunt.
In V188.8.131.52, this item was bugged on Jax to grant him maximum health when consumed.
V3.6Elixir of Fortitude is meant to be an "all-in" burst of power for players looking to take control of a situation. With its low cost, however, players were simply defaulting to it as a general starting item with no significant repercussion. Increasing the gold cost of Elixir of Fortitude aligns it better with the risky strategy it was intended for.