If I were going to pick one clip that embodied the best of Bollywood -- a clip palatable to non-fans AND avoiding outright culture-shock goofiness -- I'd pick this scene from the film "Rangeela."
What you've got here (by way of plot) is sweet bumpkin Aamir Khan trying to deal with the sudden stardom of his girlfriend, Urmila Matondkar. This song (Maghta Hai Kya) is just showing us their passion, love, unpretentious silliness, and adventurous spirit...before they begin to drift apart into different classes and ambitions.
Why's it special? First off it's a great song by unconventional musician A. R. Rahman, known for his odd rhythm, studio trickery, and long quiet stretches broken by sudden and unexpected shocks. You've also got Ram Gopal Varma directing, who at the time (1996?) considered himself the "Steven Spielberg of Bollywood," which in this case was a good thing. Finally you've got Urmilla and Aamir who had a great chemistry and a certain charm (this was during their pre-fame days...in fact, this was the film that really "made them.")
Also notable is this was when the "kiss taboo" in Indian films began to loosen: note the strange "kiss-and-cut" moment halfway through. And there's a green-screen effect that sort of works. And leaf-covered natives. And sexiness. And sweetness. You can almost excuse the traditional "standing on rocks and pinwheeling" elements near the beginning, which any fan of Bollywood has seen far too much of.
The choreography, though more adventurous than average, is still typical Bollywood: it doesn't matter if it makes sense in a cultural or traditional way, just as long as it LOOKS GOOD. Even when it doesn't. Which sums up the fun of Bollywood in general.
So in short: it's moments (and movies) like this that really MADE Bollywood for me, when I watched it regularly. Eventually I got tired of seeing the same things over and over again, and also got jaded with the craziness, but some films really shone, and "Rangeela" was one of them.