What is a non-essential clause?

A non-essential clause is a part of a sentence that is not necessary to make a thought complete; it's "decorative rather than defining."

True or False: A non-essential clause should be enclosed by commas?

True.

What is an essential clause?

An essential clause is a part of the sentence that is necessary to maintain the overall message/meaning of the sentence.

True or False: An essential clause should be enclosed in commas?

False! An essential clause needs no commas because it should be read with its subject.

Which type of clause is usually used after a proper noun (Sarah, Eiffel Tower, Dept of Agriculture?

Non-essential

The woman who sells silver jewelry is not here today.

No commas! It is essential to the meaning of the sentence.

The woman who sells silver jewelry is not here today.

Jack who loves spaghetti refused to try lasagna.

Commas! This is nonessential; it is not necessary to understand the meaning of the sentence. It's just an FYI.

Jack, who loves spaghetti, refused to try lasagna.

The book which is a Caudill had several holds on it.

Commas! It is non-essential. It's just "decorating rather than defining." 

The book, which is a Caudill, had several holds on it.

The dog that ran away has been found.

No commas! It is essential. The reader needs to know "which [dog] of many" is found.

The dog that ran away has been found.

The local diner which has amazing chicken nuggets will be closing.

Commas! The dependent clause is not necessary to understand the message.

The local diner, which has amazing chicken nuggets, will be closing.