To browse is to look casually for whatever catches your eye, rather than searching for something specific. Window shopping and scanning the newspaper for interesting headlines are forms of browsing.
You can browse in a store, a library, or browse the Internet (with — what else? — an Internet "browser"). When it comes to reading material, browse means the same as peruse. Back in the 16th century, though, this verb was used in reference to animals feeding on, or grazing on, the buds of trees.