The "Sync" command (command or ctrl-S) performs a backup and/or restore of an archive. It keeps your local copy of the archive backed up and updated with other files backed up from other computers. Files are normally downloaded to your computer as you view or export them. You can select "Aggressive Restore" in the Backup section of the Archive Settings dialog to download every file backed up from other computers during a sync, which guarantees availability for offline use.
Folder synchronization allows you to keep a folder on your computer and an archive in sync with one another. If you have multiple computers sharing a backup of the archive, it will in turn allow you to keep folders in sync across those multiple computers. To enable this, use the checkboxes for automatic import and export in the Home Folder section of Archive Settings under the Edit menu.
When you export files or folders (by selecting one or more items in List view, or the current item in Detail view), you are asked to choose a folder to export them into, and if there are files already existing in that folder you are asked whether you want to "clobber" the contents of that folder - clobbering will overwrite existing files in that folder with the same name as files you are exporting. Otherwise, BigCloud will rename the exported files and leave the existing files in the export folder untouched.
BigCloud keeps track of changes to folders, files and URLs - you can right click on any folder or file in List view to pull up the history of that location. Folder histories are only available if you have imported the entire folder, not just files within it.