The thickness of a sheet of watercolor paper is measured by weight. So, logically, the greater the weight, the thicker the sheet. It is measured either in pounds per ream (lb) or grams per square meter (gsm). The standard weights of paper are 90 lb (190 gsm), 140 lb (300 gsm), 260 lb (356 gsm), and 300 lb (638 gsm).
Thinner paper needs to be stretched to prevent it from buckling or warping when you paint on it. How thick the paper needs to be before you can happily paint away on it without buckling does depend on how wet you tend to make the paper as you paint. Experiment with different weights to see, though it's likely you'll find that paper less than 260 lb (356 gsm) wants to be stretched.
Not having to stretch it is not the only reason for using heavier paper. It'll also stand up to more abuse, and take a greater number of glazes.