Search is still the main process whereby websites get new visitors. Indeed, search generates the majority of all new traffic which is why Google is so successful.
Once someone has found a site they may bookmark it or remember it, but the first visit is usually via search. This is why Search Engine Optimisation is such a key internet marketing skill to have because unless you can get your site ranked pretty high on the major search engines, you are going to have a hard job making any progress.