Examples:

That  in a that-clause acts either as a relative pronoun or it does not.

As a relative pronoun : that  in a relative clause refers to an earlier noun antecedent, while the that (relative) clause acts as an adjective, modifying the noun antecedent.

• The beetle that  you stepped on was still alive.
• The game that  my grandparents are playing is for children.
  ( Beetle  and game  are antecedents.)

That is not a relative pronoun when it does not refer to a noun antecedent. The entire that-clause is a noun that usually follows the main verb.

• He believed that  his new business would become one of the biggest in the world .
• We discovered that  there was no truth in the rumour.
  (In both these sentences, that  is not the subject of the clause, and because it is not the subject, it can be removed from the sentence.)

• He believed his new business would become one of the biggest in the world.
• We discovered there was no truth in the rumour.   

Examples:

That-clause as subject of a sentence.

• That she could use his car tonight  was the promise he gave to her.
• That the children could fight for so long surprised the adults.

That-clause as object of a sentence.

• Mom said that we could play in the rain .
• She wished that she could sail to the Antarctic .

Examples:

That-clause follows verb. Â

• I remembered that I had left my house keys at home .
• He feared that this might be the end of his marriage .

That-clause follows adjective. Â

• The parents were excited that their daughter was getting married .
• She was annoyed that he had forgotten his wallet .

That-clause follows noun. 

• It was a mistake that he assumed his wife would agree with his plan .
• Their teacher has a firm belief that education is a waste of time .  .

Examples:

That-clause as restrictive relative clause

The that-clause is also a restrictive relative clause introduced by the relative pronoun  that . The restrictive relative clause gives important information.

• The ringmaster that the lion killed was buried yesterday. 
  
  (Without the restrictive relative clause that the lion killed , the only information we have is the ringmaster was buried yesterday. We don’t know how he died.)  
• It reminded her of the hospital that she used to work .
  
  (Without the restrictive relative clause, the question remains as to what reminded her of the hospital.)