A browser is a looker: either a program that lets you surf the Internet or a person in a store who just looks around without buying.
This is a word that has to do with looking around, otherwise known as browsing. You probably use web browsers, which allow you to see websites on a computer. But you could also be a browser if you go to a store and just look around. Being a browser is a good way to kill time, but as soon as you make a purchase, you've gone from being a browser to being a customer.