What is the minimum sized wrist the Apple Watch can fit? I am interested in getting the Apple Watch, the only problem is I have small wrists so I am not sure if the Apple Watch will fit my wrist even if I choose the 38mm (small) one.

What is the smallest sized wrist the 38mm Apple Watch can fit? (I presume the 38mm is the largest wrist the watch fits?)
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.