A user profile exists whenever a user is logged into Windows. On recent versions of Windows it is the files under
C:\Users\%USERNAME%, on older versions of Windows it was
C:\Documents and Settings\%USERNAME%. It has the documents and settings associated with a user.
A home directory is typically a network location that is automatically mapped that is setup for a single person to access.
Redirect folders is a method to have part of a person's profile like the Desktop, or Documents folder redirected to being stored on a server, or another location.
Roaming profiles is a method where a user profile is synced to a server, or some location so that a user can have the same profile for many computers.