Simply put, Let's Play (LP for short) is a video of someone playing a video game while doing commentary (usually live or post). In the vast majority of cases Let's Plays are done by individual gamers with no ties to the companies, and differs from a strategy guide or walkthrough in that it's more focused on an individual's experience with the game. Let's Plays run the gamut from humorous observations on the game to informative insight on its various aspects.
Notable Let's Play websites:
Something Awful's Let's Play subforum
Most Let's Players both well-known and obscure use YouTube, but some videos can be found on Viddler, Blip, and ScrewAttack.
Who are your favorite Let's Players? What Let's Play video do you really enjoy? Here's mine:
Chip Cheezum and General Ironicus are old and established LPers. They have very insightful commentary on the games they play, and their humorous chemistry fits well together. One of their recent series is Metal Gear Rising: Revengeance, which make for a great starting point for those new to the duo.
Slowbeef helped popularize the concept of video Let's Plays by making the first one back in 2007 (The Immortal). Although Let's Plays in screenshot format existed beforehand, many credit him with inventing the concept due to not knowing this. He works with his partner, Diabetus, and the two LP independently of one another. For newcomers, I'd recommend Diabetus' LP of Super Mario Galaxy 2.
Batman9502 tends more to the informative side of LPing, bu his videos are no less entertaining to watch. He's notable for starting out with the acclaimed Batman: Arkham Asylum game. In recent times he's played Legend of Zelda: Skyward Sword and Metal Gear Rising: Revengeance.
Table-top Let's Plays:
Yes, there are LPs of people doing table-top RPGs.
Ironicus cuts his teeth on Dungeons & Dragons in this first-ever Let's Play of Dungeons & Dragons.
Chip, Ironicus, Slowbeef, and Diabetus play a game of Dungeon World on their podcast.
Feel free to share some favorite LPers and videos of your own!