Muggles' Guide to Harry Potter/Magic/Potions

Overview
Potions is the science of combining ordinary and magical substances to create potions, (usually) liquids with magical effects.

Extended Description
Not everything discussed in Potions class is, strictly speaking, a liquid. Several sections of the class are concerned with antidotes, and on a number of occasions there is mention of a Bezoar, a stone found in a goat's stomach, which has curative properties.

There are a very large number of potions mentioned through the course of the seven books. Of these, perhaps the most important are the Wolfsbane potion, which appears in Harry Potter and the Prisoner of Azkaban, Polyjuice Potion, which Hermione first introduces us to in Harry Potter and the Chamber of Secrets, and which is then used in Harry Potter and the Goblet of Fire and Harry Potter and the Deathly Hallows, and Felix Felicis, which plays a large role in Harry Potter and the Half-Blood Prince.

Potions is taught for the first five years of Harry's schooling by Professor Snape, and in the sixth and seventh year by Professor Slughorn.

Analysis
Magic potions play a large role in Muggle ideas about magic, so it is clearly necessary that they should be a part of the Wizarding world. Their being codified and formalized into almost a science seems only proper, given what we have come to understand about them.

It is necessary for the development of Harry's character that he should not excel in all subjects, and in fact only in Defence Against the Dark Arts does Harry consistently perform to the top of his ability, when allowed to. However, particularly in Potions, Harry does not work up to the level that we expect of him. Certainly, some of this is due to unfair marking of assignments by Snape, who we see as having a bias against Gryffindor house in general, and against Harry in particular.

"...and was pleased to find that when left alone, he was able to concoct an Invigoration Draught quite easily." —Harry's opinion on how easy it was to brew the OWL-level potion

Only later in Harry's magical education does his true ability at Potions show when he is only one of twelve students able to make it to NEWT-level, having achieved an "Exceeds Expectations" in his OWL examination. It is noted that Harry has fewer problems with potions as long as Snape is not around.

Greater Picture
For the first five years, Potions is inextricably tied up with Professor Snape, and Harry's performance in that course is clearly influenced by Snape's dislike of Harry. A large part of Harry's difficulty with Potions lies in Snape's trying to force Harry into the same mold as his father, and Harry's unconscious resistance to that. In one particular case, we see that Harry's failure to correctly make a potion is due to Snape's having written the instructions on the board in a way that would cause confusion in any reader. It is clear to the reader that Snape is trying to make Harry fail, and does not care if he dooms the entire Gryffindor class at the same time.

It is noteworthy that, given a fair teacher (Slughorn) and a good instructor (the "Half-Blood Prince"'s book), Harry is capable of turning in an excellent job in Potions class, and even absent the book he is still able to do a decent enough job of it that Slughorn's suspicions are not raised. It should also perhaps be noted that Harry's mother is singled out by Slughorn as one of the best Potion students he had ever had.