That is omitted in the Attribute Clause and object clause.

That guideAttribute ClauseThat can be omitted when the clause is an object. (That is a relational pronoun)

That inObject clauseCan be omitted. (That is a conjunction)

However

That of the guiding object clause is generally not omitted in the following cases:

1.There is an insertion before the object clause. For example:

We hope, on the contrary, that he will stay at home with us.
On the contrary, we want him to stay at home with us.

2.When indirect objects exist. For example:

He told me that he was leaving for Japan. He told me that he is going to Japan.

3.That is after another object clause. For example:

He said (that) the book was very interesting and that (not omitted) All the children like to read it. He said the book was interesting and all the children liked to read it.

4.After "It (formal object) + complement. For example:

I think it necessary that he shoshould stay here. We think it is necessary for him to stay here.

5.That clause is used to answer questions separately. For example:

-What did he hear?
What did he hear?

-That Kate had passed the exam. (He heard) Kate passed the test.

6.After T. For example:

He has no special fault doesn't that he smokes too much. He has no special problem except smoking too much.

7.At the beginning of a sentence. For example:

That our team will win, I believe. I believe our team will win.

8.After a formal or uncommon verb (such as reply or object. For example:

He replied that he disagreed. He replied that he did not agree.

Reference: http://www.yygrammar.com/Article/200809/489.html