Relative Clauses

There are two different types of relative clause:

A "defining" or identifying clause, which tells us which person or thing we are talking about.
A "non-defining" or non-essential clause, which gives us more information about the person or thing we are talking about. This kind of clause could often be information included in brackets (...)

Example:

The farmer (his name was Fred) sold us some potatoes. The farmer, whose name was Fred, sold us some potatoes.

It is important to see the difference between the two types of clause, as it affects:

a. the choice of pronoun used to introduce the clause,
b. the punctuation - you must use commas with a non-defining clause.

http://www.edufind.com/english/grammar/get_alpha.cfm?letter=R

When can I leave out who, which, and that in relative clauses?

Sentences can be divided into parts called clauses. A relative clause is a part of a sentence that describes the person or thing we are talking about and is connected to other clauses in the sentence via a ‘relative pronoun’, who, which or that. Relative pronouns replace the subject or object of the verb:
Where is the new boy? He was in class yesterday.
Where is the new boy that was in class yesterday?
Can I borrow the CD? You bought the CD.
Can I borrow the CD that you bought?

identifying relative clauses

The girl that I sit next to in class, gave me her phone number.
that I sit next to in class (the relative clause) identifies ‘which’ girl (there could be many girls in the class).

non-identifying relative clauses

This is my friend, Thomas, who came on holiday with me last year.
who came on holiday with me last year does not identify ‘which’ friend (we know ‘which’ friend - the friend is Thomas).

Note! In written English, non-identifying relative clauses are separated by commas, and in speech, by pauses.

keeping who, which and that

You cannot leave out who, which, that when:
it is the subject of the verb in the relative clause
Where is the new boy that was in class yesterday?
it is part of a non-identifying relative clause
The trees that at one time lined this road have all been cut down.

leaving out who, which and that

You can leave out who, which, that when:
it replaces the object of the verb in the relative clause

Can I borrow the CD (that) you bought?

Note! In identifying relative clauses, where which thing or person talked about is clear without the relative clause, it is very common in spoken English to leave out who, which, that.

http://www.britishcouncil.org/learnenglish-central-grammar-definitions-relative-clauses.htm