The conjunction that is often left out, especially in an informal style. Here are the details.

Indirect speech

That can be left out after common reporting verbs such as say, think, suggest etc.

Alice said that she would come. OR Alice said she would come.
She suggested that I should consult a doctor. OR She suggested I should consult a doctor.

That cannot be dropped after certain verbs. Examples are: reply and shout.

James replied that he had finished the work. (NOT James replied he had finished the work.)
She shouted that she was going. (NOT She shouted she was going.)

After adjectives

Some adjectives can be followed by that-clauses. That is left out in common expressions.

I am glad you have come. OR I am glad that you have come.
We were surprised that he phoned. OR We were surprised he phoned.

Conjunctions

Some common two word conjunctions (e.g. so that, such…that, provided that, supposing that, considering that etc.) are often used without that.

She walked in quietly so that nobody would hear her. OR She walked in quietly so nobody would hear her.
Assuming (that) we win the first prize, we will throw a party next week.
I have such regard for him (that) I will do anything to make him happy.

Relative structures

The relative pronoun that can be left out when it is the object in a relative clause.

These are the people that we met at the club. OR These are the people we met at the club.