What is Mac's equivalent to Windows Explorer (file manager not internet explorer)?

It's called Finder and yes you can copy your documents to an external drive. The drive icons on the side of your desktop invoke finder windows. Inside are folders which can be displayed with details or icons. You can drag and drop just like windows.

If you don't have drive icons on your desktop then click anywhere on the desktop and look at the top menu bar. It will say "finder", now use the menu to select finder then preferences and turn on the icons. You should also have a square blue "smile" icon on the dock which invokes finder.

Finder will work, but for backup, consider using Time Machine to back up all files, (Included with OSX) or Carbon Copy Cloner to backup selected files (a 3rd Party App, not free).

See also: Moving from Windows Explorer to the Finder

and : http://www.apple.com/support/macbasics/pctomac/

Yes, use Apple's Time Machine software to automatically copy everything to your external drive. It will then maintain a backup of whatever has changed every hour. It is essential. Don't spend time manually doing backups!

I don't think you can backup all your documents to iCoud, even if you wanted to.