What is the minimum sized wrist the Apple Watch can fit?
As specified on Apple's Watch Case and Band Sizes page, the smallest band (therefore fitting the smallest size wrist) is the Classic Buckle, which fits a 125mm wrist circumference.
The next size of bands, are the ones that fit 130mm wrist circumferences:

Milanese Loop (130-180mm)
Classic Buckle (125-200mm)
Sport Band (130-180mm and 150-200mm)
Woven Nylon (125-195mm)

Keep in mind these are sizes for the 38mm Apple Watch. 42mm band sizes are a bit larger, starting at 140mm for the Link Bracelet and Sport Band, and 145mm for Woven Nylon.