Chrome is google's web browser. It is the only browser available on chromebooks, but is the suggested browser for use most of the time when working with google products (e.g. google docs, google slides, etc.)
When using a Chromebook, you automatically log in when logging on to the machine. This makes your bookmarks and history visible on that machine. When you log out of the chromebook, all of that information is not accessible to others.
On other platforms (e.g. Macbooks): Logging In and Out
Bookmarks & Teacher Websites - guide for organizing bookmarks and finding teacher websites
Session Buddy - Chrome extension that can help organize tabs or recover lost tabs after a crash
Chrome Bookmarks - watch this 2 minute video on how to save and organize bookmarks