In the 1980s, Jim Carrey was in A Duck Factory, a TV series focusing on the life of a cartoonist.
employs the active form of the verb,
In the 1980s, Jim Carrey was in A Duck Factory, a TV series (that was) focused on the life of a cartoonist.
employs the passive form (with, as is common, ellipsis of relative pronoun and auxiliary).
The verb 'star' on the other hand, works quite differently. It has two active meanings,
(1) as a transitive verb, meaning 'contain/feature', with a film/show, etc. as subject and a famous person as object, e.g.
The show starred Jim Carrey.
(2) as an intransitive verb, meaning 'appear', with the famous person as subject and the show/film indicated by a prepositional complement, e.g.
Jim Carrey starred in the show.
(Interestingly, 'star' (1) is one of a relatively small number of transitive verbs that have no corresponding passive. Thus we cannot transform the above to
*Jim Carrey was starred by the show.)
To turn now to the earlier sentences,
The Fresh Prince of Bel-Air was a popular 1990s sitcom starring Will Smith.
works, of course, because it employs the active (present) participle of the transitive verb, whereas
*The Fresh Prince of Bel-Air was a popular 1990s sitcom starred Will Smith.
does not, since we cannot perform the transformation that worked with regard to the Jim Carrey sentence, which would yield only nonsensical
*The Fresh Prince of Bel-Air was a popular 1990s sitcom that was starred Will Smith.
And since, as noted above, no passivization of transitive 'star' is possible, even the insertion of 'by'
*The Fresh Prince of Bel-Air was a popular 1990s sitcom that was starred by Will Smith.
would fail to improve matters.
I trust that makes things a little clearer!