When can you exclude "that" from a sentence?

There are a number of instances in English where it is possible and even desirable to omit that.

• that as conjunction with reporting verbs (like learn, discover, find out, know, feel, etc.): I discovered (that) Julian had borrowed my car without my permission.

• after the more common reporting verbs, (e.g. say, tell) it is also entirely natural to omit that in informal speech: I told him (that) I'd be back by ten o'clock but he said he needed me here by nine.

• after certain verbs (e.g. reply, shout) that cannot be omitted and it is not normally dropped after nouns: The Dean of the Humanities Faculty informed the students that the drama dept was going to close.
  
  Often, that can be removed from a sentence without affecting the overall meaning, especially when it has been used with a conditional, e.g. She told me that I could go to the party vs. She told me I could go to the party.

Sometimes, however, you can change or lose the meaning by removing that. For example, I heard that you snore = Person A snores, and Person B has told Person C about it, whereas I heard you snore = Person A snores and Person B has heard them doing it.

Consider whether your use of "that" is necessary for understanding time, object or person. If not, you can probably cut it.