The assistant for a user is an email address (or set of email addresses, separated by semicolons) where incoming messages to this user will be copied or redirected.
Normally messages will be copied to the assistant, but if the Redirect to assistant (don't keep a local copy) option is checked, then the message will just be sent to the assisant, and not go into this user's mailbox as well.
An important difference between an Assistant and a normal forward is that messages FROM the Assistant themselves are not forwarded back to the Assistant.
So, if user A has an assistant B, with 'redirect to assistant' enabled, then if someone else sends a message to A, B will get the message, not A. But then, if B thinks that the message is important, and forwards it back to A, because the message came from the Assistant, the message will not be forwarded back to B, and will actually go to A - ignoring the assistant setting.