Elastic waistband is a band of stretchy material, that is sewn into the waistline of a garment (like pants, skirts, shorts, pajamas, etc.). Elastic waistband primary purpose is to provide a comfortable and adjustable fit around the wearer's waist.

The core of an elastic waistband is the elastic itself, which can be made from rubber, latex, spandex, or a blend of these and other fibers. This allows the waistband to expand and contract.
The elastic band is usually sewn inside a folded-over portion of the garment's fabric (creating a casing) or attached to a separate fabric band that then becomes the waistband. This protects the elastic and provides a more finished look. Sometimes, the elastic can also be directly sewn onto the fabric.   

The elasticity allows the waistband to conform to the wearer's body without being too tight or restrictive. It also accommodates slight changes in waist size.   
While not as precisely adjustable as a drawstring or belt, an elastic waistband allows for some flexibility in how the garment fits.
Garments with elastic waistbands are often easy to pull on and off, making them convenient for everyday wear, children's clothing, and loungewear.