"Ozzie and Harriet" is often used as a buzzword for white-bread America: Husband runs the family spouting words of manly wisdom, while the wife stays home with the well-behaved kids. Funny thing is, the show really isn't like that. Ozzie is a guy who apparently never goes to work - it's a running gag throughout the show. His "great ideas" usually lead to disaster, and usually it's Harriet who quietly gets everything to turn out all right in the end. The kids, especially Ricky, often shoot off at the mouth. It was even Seinfeld-esque (and I say that as a rabid Seinfeld fan) - most episodes could fairly be described as being "about nothing".
In truth it's one of the funniest shows ever on television. It was even cutting edge, for its time: Ozzie and Harriet slept in the same bed, which was unheard of. Ever see anyone on a TV show "break the fourth wall" (start talking to the camera)? This started on O&H - first with Ricky's end-of-show shrugs, and later with full-blown conversations directed to the camera. My personal favorite example of this is when Ozzie pretended to be a mind-reader (who of course no one recognized because of a cheesy goatee). When he gets exposed at the end, just about every character quips something or other straight into the camera.
Do yourself a favor though. Don't start off with the late episodes where the boys are grown up and married. Those can be quite funny, but the show at times was just coasting on its reputation by then. Watch the earlier stuff from when the boys were little, when Thorny still lived next door. Give yourself time to get to know the characters, and you certainly won't regret it!