This article is from the alt.usage.english FAQ, by Mark Israel email@example.com with numerous contributions by others.
In "The family that prays together stays together", the clause
"that prays together" is called a RESTRICTIVE CLAUSE because it
restricts the main statement to a limited class of family. In
"The family, which is the basic unit of human society, is
weakening", "which ... society" is called a NONRESTRICTIVE CLAUSE
because it makes an additional assertion about the family without
restricting the main statement.
It is generally agreed that nonrestrictive clauses should be
set off by commas; restrictive clauses, not. Nonrestrictive
clauses are now nearly always introduced by "which" or "who"
(although "that" was common in earlier centuries). Fowler
encourages us to introduce restrictive clauses with "that"; but this
is not a binding rule (although some copy-editors do go on "which
hunts"), and indeed is not possible if a preposition is to precede
the relative pronoun. "Which" seem to have more "weight" than
"that"; the weight often just adds starch, but it can be of use
when the relative pronoun is separated from the antecedent: "This
is the only book in my personal library which I haven't read."
Often, too, euphony favours one or the other.
Object relative pronouns can be omitted altogether ("the book
that I read" or "the book I read"); in standard English, subject
relative pronouns cannot be omitted, although in some varieties
of informal spoken English, they are ("There's a man came into
the office the other day").
Robert Sigley (Robert.Sigley@vuw.ac.nz) is writing a Ph.D.
thesis on relative pronoun choice.