"One simply siphons the excess thoughts from one's mind, pours them into the basin, and examines them at one's leisure. It becomes easier to spot patterns and links, you understand, when they are in this form."
-- Dumbledore (GF30)
A Pensieve is a shallow stone basin with ancient Saxon runes and symbols carved around the edge used to collect and view memories (GF30, Pm). When in use, a silvery light shines from its contents, which are bright, whitish silver, and cloud-like, moving ceaselessly. Harry thought the stuff in the basin looked like “light made liquid – or like wind made solid” (GF30).
Pensieves are very rare and powerful magical items. They are intensely personal as well, since memories often hold recollections of things one would wish to hide or forget. The Pensieve of a powerful witch or wizard is typically buried with them when they die. However, the Hogwarts Pensieve does not belong to a particular person but to the school. It contains centuries of magical memories left behind by previous Heads and serves as a magical reference source. The Headmaster or Headmistress literally has access to the wisdom of the ages.
Legend has it that the Hogwarts Pensieve was found half buried in the ground on the very spot where the Founders decided to build their school (Pm).
Dumbledore kept his Pensieve in a black cabinet in his office. When he had too many thoughts and memories crowded into his head, he siphoned the excess thoughts into the basin using his wand. He said that "(i)t becomes easier to spot patterns and links" in the memories when they're collected in the Pensieve (GF30).
Harry touched the material in Dumbledore's Pensieve when he first saw it, and was taken back incorporeally into some of Dumbledore's memories (GF30).
Snape borrowed Dumbledore's Pensieve early in 1996 to prepare his mind for Occlumency lessons with Harry. Snape wanted to remove some of his more embarrassing memories so that Harry would have no chance of seeing them (OP24). Unfortunately, Snape left the Pensieve on his desk, and when Snape was out of the room Harry visited one of those memories (OP28).
During Harry's sixth year, Dumbledore used memories he had collected from various people and stored in small crystal bottles to give Harry lessons about the life of Tom Riddle/Lord Voldemort. Each memory was poured into the Pensieve, then Dumbledore and Harry entered as bystanders to the action (HBP10, HBP15, HBP17, HBP20, HBP23) .
Horace Slughorn was ashamed that he had given Tom Riddle information about how to make a Horcrux, so he tampered with the memory to cast himself in a better light. That memory appeared to have "congealed" as if it had "gone bad" in the bottle, and there was a strange fog over the scene (HBP17). With the help of the potion Felix Felicis, Harry was able to prey on Slughorn's guilt and persuade him to hand over the real memory (HBP22, HBP23).
The first two times Harry watched Memories in the Pensieve, he intruded out of curiosity and saw more than he bargained for. Dumbledore chided him for it later, saying that during his lessons he would give him "permission" to watch the memories (HBP15).
J.K. Rowling has explained how the Pensieve works in several interviews:
Q: Do the memories stored in a Pensieve reflect reality or the views of the person they belong to?
A: It's reality. It's important that I have got that across, because Slughorn gave Dumbledore this pathetic cut-and-paste memory. He didn't want to give the real thing, and he very obviously patched it up and cobbled it together. So, what you remember is accurate in the Pensieve.
Q: So there are things in there that you haven't noticed personally, but you can go and see yourself?
A: Yes, and that's the magic of the Pensieve, that's what brings it alive.
Q: I want one of those!
A: Yeah. Otherwise it really would just be like a diary, wouldn't it? Confined to what you remember. But the Pensieve recreates a moment for you, so you could go into your own memory and relive things that you didn't notice at the time. It's somewhere in your head, which I'm sure it is, in all of our brains. I'm sure if you could access it, things that you don't know you remember are all in there somewhere.
The name "Pensieve" is a play on words. In the first place, "Pensieve" is a homonym for the English word pensive. In the second place, the word "Pensieve" is formed from the Latin pens + English sieve.