Dysfunction Junction: Larry's family seems like a bit of a mess. Anytime he even mentions his family, he talks about "getting hollered at", and they openly say that his father hits him and his sister. His father seems to travel frequently and every time he's out of town, his mother acts like a basket case. There are also mentions of her getting nervous headaches when her children misbehave AND when they get disciplined. It's treated as a joke, but the longer you watch, the sorrier you feel for Larry. He's not a very bright kid but his home sounds like an unpleasant place, which may explain why he spends so much time at Beaver's house (something Wally comments on fairly often, 'gee, with the number of meals he's eaten here...' etc.)
Indeed, Tony Dow ended up becoming so popular, that in the later seasons, the writers started writing episodes in which Wally would get into the schemes and scrapes that Beaver would usually get into, with Beaver becoming more of an observational character.
And, of course, Eddie Haskell.
Not to mention Clarence "Lumpy" Rutherford, bad girl Judy Hensler and even Gilbert Bates.
Hilarious in Hindsight: In one episode everyone was aghast that Beaver was actually selling water to people! Mind you, it wasn't so much that he took advantage of inside knowledge that the water main would be shut off or that he was selling it out of a bucket on his wagon, but merely the fact that one person could be low enough to sell water to another person! How Evian spelled backwards of them!
Keep Circulating the Tapes: For The New Leave It to Beaver series. Due to legal issues, the revival series has never been released on DVD.
Unintentional Period Piece: Leave It To Beaver is what pretty much everybody (who wasn't there) thinks middle-class suburban life was like in the 1950s. The book The Way We Never Were by Stephanie Coontz begins with a takedown of shows like this: "Contrary to popular opinion, Leave it to Beaver was not a documentary."