Trying to understand the history behind Russell's Paradox

[Dr H]

Some entities in the domain would be mapped by f on to subclasses that contain them, whereas others may not.

Can anyone explain what the author means here, especially the last part of that sentence? What "others" may not be mapped by f on to subclasses that contain them? How in this context can an entity be mapped to a subclass not containing it? Or am I misunderstanding the term "map" here?

I want to try to understand this bit by bit so I can fully understand the whole paragraph. Thanks.

Russell's Paradox is a paradox of self-reference that leads to essentially a circular reasoning problem.

The "others" in the problematic statement are the "entities" which either belong to particular subclasses, or not.  
That classes/subclasses themselves may also be "entities" is where the problem arises.

For example:  "The set of all bowling balls" defines a limited set of entities -- bowling balls.  

"The set of all things that are not bowling balls" defines a much broader set of entities -- everything which is not a bowling ball.  This would necessarily include the set itself, since "the set of all things that are not bowling balls" is itself, clearly not a bowling ball.  (It would also include the set of all bowling balls -- just the set; not the balls --, since the set, as an entity, is not itself a bowling ball.)

So the second set includes itself as an element of itself, whereas the first set does not.

Don't know if that helps you or not.  As an engineer I often find that viewing things in concrete terms makes them more graspable, but the same can't always be said for others.