A cleanser removes everything your skin collects throughout the day: sweat, sebum (your natural oil), dead skin cells, makeup, SPF, pollution, and dirt. Without cleansing properly, all of that sits on your skin and blocks your other skincare from absorbing.
A good cleanser does its job without stripping your skin barrier. You want it to leave your face feeling clean but not squeaky or tight. There are two main types: oil cleansers (great for dissolving makeup, SPF, and excess sebum) and water-based cleansers (foam, gel, micellar, all great for everyday use).
The Korean approach is to use both, layered. Oil cleanser first, water cleanser second. This is called double cleansing and is especially important if you wear SPF or makeup daily.