Felix Felicis is a potion, otherwise known as "liquid luck" as it makes the person who drinks it extremely lucky for a certain amount of time.
When taken it causes feelings of exhilaration and confidence, giving the one who has taken the potion a sense of what they should do and it feeling 'right'. It is powerful enough to be a banned substance in organised competitions, such as examinations, sports, or elections.
If taken in excess it has negative consequences. It will cause giddiness, recklessness and overconfidence.
Only a little bit of potion will give you good luck. You do not need to take absurd quantities of it, or the above side effects will occur. This potion is also highly toxic if taken in large quantities.
 Sixth Book, the Half-Blood Prince
In 1996 during Harry Potter's first Potions lesson with Horace Slughorn, he chooses to open the lesson with an impressive display of tricky potions which they should be able to brew by the end of studying for the N.E.W.T.s. One of these is Felix Felicis, which Slughorn offered as a prize to the student who made the best first attempt at the Draught of Living Death. This student turns out to be Harry himself, although this is due to his copying the handwritten directions left by the mysterious 'Half-Blood Prince' in his borrowed copy of Advanced Potion-Making.
Slughorn warned the class that Felix Felicis is banned in organised competitions, so when Harry faced the prospect of captaining a Quidditch team to defeat, in part because of his best friend Ron's lack of confidence as a player, he does not give in to temptation and give him the potion. However he successfully tricks Ron into believing that he has, when their friend Hermione believes she sees him doing just that. This sudden misplaced boost of confidence was all Ron needed to play well and the Gryffindor Team won the match.
The potion was used in part by Harry Potter, when he wished to extract an important memory from Professor Horace Slughorn under the instruction of Albus Dumbledore. The rest of the potion was used during the Battle of the Astronomy Tower, when Harry donates the potion to his friends to protect them from the attack of the Death Eaters
 Making the Potion
The potion within was splashing about merrily; it was the colour of molten gold, and large drops were leaping like goldfish above the surface, though not a particle had spilled.
Felix Felicis is an advanced potion. It takes sixth months to make and according to Slughorn is disastrous if you get it wrong. This difficulty is what hinders Hermione Granger from making the potion to help Harry Potter and Ronald Weasley in their mission against Voldemort.
 Additional Canon
 The Video Game
During the time when Harry takes the potion, the gamer is able to get the maximum Mini-Crests out of every object. Therefore, it is not difficult to get an easy one or two Crests during this time. However, the gamer has no control over the character; Harry moves on his own, and at a rapid speed, so you must be quick while getting the Mini-Crests!
 The Films
However, the movie version of the potion differes greatly from the books. In the books, the potion is "molten gold" but the movie showed the potion as a clear, watery liquid.