The Leaky Cauldron is a pub and inn in Diagon Alley, run by a bald wizard named Tom.
The Leaky Cauldron is actually the barrier through which wizards can get from Muggle London to Diagon Alley. The pub has a spell cast on it so that Muggles don't notice the pub and walk in without knowing what they're getting into.
To get into Diagon Alley, all wizards have to do is go out to a small, walled courtyard behind the pub. Find a trash can, and count three bricks up and two across. Tap that brick three times, and abracadabra! Welcome to Diagon Alley
What do you do here?
Well, it's a pub! So, you buy drinks! Well, drinks and food. Yum. Also, there are rooms to use. A perfect getaway for any traveling wizarding family.
What does this place have to do with Harry?
Book 1: When Harry goes to the Leaky Cauldron, he realizes for the first time that he's famous. Tom and the people at the bar all shake hands with Harry. Here he also first meets Professor Quirrell, his first (of many) Defense Against the Dark Arts teachers.
Book 2: Harry leaves Diagon Alley through the Floo at the pub.
Book 3: After Harry leaves the Dursleys via the Knight Bus, Fudge is waiting for him at the Leaky Cauldron. There, Harry finds out that something odd is going on (I mean, he blows his aunt up like a balloon and nothing happens? Come on...) and stays there until going back to Hogwarts. Also there, he overhears Mr. and Mrs. Weasley talking and realizes that Sirius Black is after him.
Book 4: There's no excitement in the Leaky Cauldron in this book.