Yarn weight refers to the thickness of the yarn, not how heavy it is. The Craft Yarn Council standardized this into categories 0 through 7.
Category 0 is lace weight, the thinnest. Category 1 is fingering or sock weight. Category 2 is sport. Category 3 is DK or light worsted. Category 4 is worsted or medium, the most common. Category 5 is bulky. Category 6 is super bulky. Category 7 is jumbo.
The weight determines your hook or needle size, your gauge, and how much yardage you'll need for a project. Thicker yarn uses fewer yards but works up faster. Thinner yarn uses more yards but creates finer detail.
You can identify yarn weight by checking the label, or by wrapping it around a ruler and counting wraps per inch (WPI). More wraps means thinner yarn.