Muggles' Guide to Harry Potter/Magic/Pensieve

Overview
The Pensieve is primarily a device for storing memories outside of one's own head.

Extended Description
The Pensieve has multiple functions.

At times, when one's head is so full of thoughts that one cannot hear oneself think, it is useful to be able to take some of those thoughts and literally set them aside. The practiced Wizard can extract a thought from his head and store it in a phial or in the Pensieve for another time. If it is in the Pensieve, it is possible to stir the thoughts stored there together and look for patterns. It appears that the wizard has the choice of extracting an entire memory, leaving no trace of it in his head, as Professor Snape does in Harry Potter and the Order of the Phoenix, or extracting a copy of a memory, retaining the original, as Professor Slughorn does in Harry Potter and the Half-Blood Prince. It is also apparently possible to edit these extracted memories, though it is a difficult task and one which is often not done well.

If one places one's head within the Pensieve, one becomes immersed in a memory that is stored in the Pensieve, and is able to relive it as if one was living that time over again. Harry experienced Professor Dumbledore's memories of the Wizengamot trials of several Death Eaters this way in Harry Potter and the Goblet of Fire, and Professor Snape's memories of Harry's father in Harry Potter and the Order of the Phoenix.

A thought or memory stored in the Pensieve can, with proper stimulus, appear to nearby viewers as if standing on the surface of the basin. Professor Dumbledore used this technique to show Harry the prophecy that had been made about him, in Harry Potter and the Order of the Phoenix, and it is used in Harry Potter and the Half-Blood Prince when full immersion in memory was not needed.

It is also possible to take another person's memories, place them in the Pensieve, and then enter them to relive them as if one were the person whose memories you have just added to the Pensieve. Harry and Professor Dumbledore do this a number of times in Harry Potter and the Half-Blood Prince in order to determine the salient points of the early history of Tom Riddle, or as he later styled himself, Lord Voldemort.

Analysis
Most interestingly, the memories viewed by the person watching in the Pensieve are more complete than the person's own observations. For instance, in Harry Potter and the Half-Blood Prince, Bob Ogden visits the Gaunt family. Morfin speaks to him only in Parseltongue, which Ogden does not know and so would not be able to remember properly; yet Harry, reliving Ogden's memories, not only understands what the Gaunts are saying in Parseltongue, he is able to perceive things happening outside Ogden's range of vision.

Greater Picture
In Harry Potter and the Half-Blood Prince, a number of times we are privileged to see the memories of wizards long dead – Morfin Gaunt and the house-elf Hokey, to give two examples. From this, we can be certain that extracted memories, if preserved in vials, live on past the death of their owners. Dumbledore's Pensieve, whenever we see it, is swimming with memories that Dumbledore has set aside; we never do find out if Dumbledore's memories in the Pensieve survive his death. If they did survive, it would seem extremely likely that Harry would have need to consult these memories in Harry Potter and the Deathly Hallows, but the situation does not arise. He does use the Pensieve, however, to review the memories of another dead wizard, namely Snape.