Does anyone know if my laptop will store every website I go on from the time I start using it?
You mean from the time you first using your computer, regardless of old it is, right? No, it does not endlessly continue storing websites. The length of time depends on the settings in your web browser. In my Windows 7 computer, Internet Explorer is set to keep up to 20 days of history or up to 50MB of disk space, whichever happens first. You can change these settings. In Internet Explorer, go to Tools, Internet Options, select the General tab, and then click in Settings (under Browsing history). If you're using some other browser such as Firefox, Chrome, etc., the procedure would be similar but the menu structure and screen views will be different from Internet Explorer.