I've never heard of kettlebells until the last couple of years. It seems that they've kind of exploded recently, and a ton of people are extolling the virtues of these things. I've heard that they were popular in Eastern Europe for a long time, and have recently broken out to the rest of the world. My question is why are they so popular? What can they do that a regular dumbbell can't?
The kettlebell craze is not about the tool, it's about the movements.
The most basic movement with a kettlebell is swinging it between your legs, both as an exercise of its own, the kettlebell swing, and to enable high repetition full body exercises to get some really good cardio during your resistance training. Also, this movement is excellent training for your posterior chain, giving kettlebells a reputation to produce buttocks of steel and healthy backs.
Kettlebell Exercises tend to build functional strength by using many muscles together, not pumping up single muscles by isolating them. The fact that the center of gravity of the weight is not in your hand, requiring that extra effort to stabilize the bell, also gets more muscles involved.
If you're not interested in the kettlebell school of movement, you don't need kettlebells. Here are some example videos of what it looks like.
Concerning timing, the reason kettlebells have been getting popular these last years is that Pavel Tsatsouline, the man who started it, did so in 1998 by writing a article called "Vodka, Pickle Juice, Kettlebell Lifting, and Other Russian Pastimes", later founding the RKC organisation, training kettlebell instructors, producing kettlebells in the US and so on. Kettlebells have been popular in Russia for a long time, Pavel brought them to the US.
The quick spread and breakout of popularity has to do with the general mindset shift of Western training to a more Eastern training and fitness. In the West people have always trained to "look good" by isolating certain muscles i.e. bicep curls and machines that focus on a specific muscle, in contrast a Eastern regime is more functionally geared and involves compound movements with the use of kettlebells. Kettlebells DO NOT isolate muscles unlike a traditional dumbbell. IMO