Reported speech can also be called an indirect quotation. When we relay what another person has said, we usually don't use his or her exact words. What we use is reported speech. When describing reported speech, or an indirect quotation, it is helpful to contrast it to a direct quotation.
In a direct quotation, the speaker or writer is quoting word-for-word. When punctuating for a direct quotation in writing, the words must be written exactly as spoken inside quotation marks.
ex. "I will not give a single test this year," the math teacher said.
An indirect quotation does not need quotation marks. It can be used to say an approximation of what another person says, thinks, or believes.
ex. He said that he would not give a test this year.
When using an indirect quotation/reported speech, it is a finer point of grammar to always use "said that" before the content you are relaying. "He said" should be followed by an exact, direct quotation, but "he said that" can be a restatement.
ex. He said, "I will not give a single test this year."
He said that he would not give a test this year.