Two strokes make more power than 4 strokes, but the torque is higher in the power band. Additionally, properly maintained modern 2-strokes are NOT the polluters as in the past, especially new. Not clean, mind you, but not the smoke belching beasts of the past. They are popular because they are cheaper, MUCH easier to maintain (although a shorter life), and have more oomph than a 4-stroke. When you're talking 50-125cc's, more oomph is big. Their price/performance ratio is high, just as the Honda/Yam/Suzuki price/quality ratio is high.
The quality gap between the modern Chinese knock-offs and the Konda/Yam/Suzuki lines is closing quickly. Honda/Yam/Suzuki are having to find cheaper manufacturing methids to compete, and the knock-offs are getting much better quality from their parts. 3-5 years ago the quality gap and parts availability was large. Today, not so much.
That said, the Honda Cub 50/70/90 series, yes, those ugly utiltarian motos in dull colors, are the largest mass-produced vehicles in the history of the world. THEY are the ones you see, beat to crap, and still ticking. You cannot go wrong with those.