I haven't read the books, but I have watched the movies many times already. If this scene is in the books, I would like to know the reference (i.e. the page and the version of the book). In The Return of the King, when Aragorn, Gandalf the White, Legolas, Gimli, Eomer, Pippin, Merry, Gondor, and Rohan soldiers came to the Black Gate to distract Sauron from Frodo and Sam, the gate opened and the Mouth of Sauron rode out on horseback. My question is, Why didn't the Mouth of Sauron just lead the orc army to fight Gondor and Rohan instead of talking to them? He is the lieutenant of Barad-dûr.
The Mouth of Sauron was the Lieutenant of Barad-dûr, and therefore presumably under the direct command of Sauron. His job at the beginning, according to Sauron's plan, was first to give the Army of the West hope (by showing them that Sauron might treat with them), then to crush their hope by showing them how hopeless their situation was, and finally to crush their army:
There was a long silence, and from wall and gate no cry or sound was heard in answer. But Sauron had already laid his plans, and he had a mind first to play these mice cruelly before he struck to kill. So it was that, even as the Captains were about to turn away, the silence was broken suddenly. There came a long rolling of great drums like thunder in the mountains, and then a braying of horns that shook the very stones and stunned men’s ears. And thereupon the door of the Black Gate was thrown open with a great clang, and out of it there came an embassy from the Dark Tower.
An embassy, in this usage, means a mission or a group of persons on a mission to conduct diplomacy. And indeed this is what the Mouth of Sauron pretends to do:
Aragorn said naught in answer, but he took the other’s eye and held it, and for a moment they strove thus; but soon, though Aragorn did not stir nor move hand to weapon, the other quailed and gave back as if menaced with a blow. 'I am a herald and ambassador, and may not be assailed!' he cried. ... Then thou art the spokesman, old greybeard? ... I have tokens that I was bidden to show to thee—to thee in especial, if thou shouldst dare to come.
This then is the answer: The Mouth of Sauron reported directly to Sauron himself, and Sauron in his malice wanted to play mind games with the leaders of the Army before he wiped them off the face of the earth. The Mouth, for his part, considered himself a herald rather than a part of the military, and appears not to have had the mind (or the bravery) for fighting. (Thanks to NKCampbell for noticing this quote!)
But he [the Mouth of Sauron] looked at the fell faces of the Captains and their deadly eyes, and fear overcame his wrath. He gave a great cry, and turned, leaped upon his steed, and with his company galloped madly back to Cirith Gorgor.
(All quotes from The Lord of the Rings, Book VI, Chapter 10, "The Black Gate Opens"; all emphases added)
The Mouth of Sauron appears in
Return of the King Chapter 10 - The Black Gate Opens
Without knowing the exact format and edition of your book, it would be impossible to give a page number
Who is the Mouth of Sauron? The same chapter tells some of his background:
But it is told that he was a renegade, who came from the race of those that are named the Black Numenoreans; for they established their dwellings in Middle-Earth during the years of Sauron's domination, and they worshipped him, being enamored of evil knowledge. And he entered the service of the Dark Tower when it first arose again, and because of his cunning...he learned great sorcery
As to why he didn't fight - one can only speculate, however:
"But he [the Mouth of Sauron] looked at the fell faces of the Captains and their deadly eyes, and fear overcame his wrath. He gave a great cry, and turned, leaped upon his steed, and with his company galloped madly back to Cirith Gorgor. But as they went his soldiers blew their horns in signal long arranged; and even before they came to the gate Sauron sprang his trap.
Given the little that is told in the book, we can presume then that:
1) he was afraid of the leaders of the company that came to challenge Sauron and fled. Additionally, this seemed to be part of a plan (see the text: "long arranged, sprang his trap")
2) the Mouth of Sauron was deep in the service and mind of Sauron, and, being great in sorcery, may not be inclined to sully himself with sword play
Remember the whole point of marching on the Black Gate was to try and trick Sauron into thinking Aragorn or Gandalf held the one ring. The mouth of Sauron was therefore initially cautious, trying to determine what secret weapon the heroes had to hand before springing his trap. If one of them had wielded the ring and he had simply moved out to attack he could well have been beaten allowing the heroes to march on Sauron.
Sauron never considered that they might wish to destroy the ring believing instead that Man would always be tempted to wield the great power. This was also compounded by the fact that when Pippin gazed into the palintir of Orthanc Sauron became convinced Saruman had found the one ring. When his riders got to the tower and found it destroyed he then believed that the forces of good had now taken possession of the ring and where taking it to Gondor to use against him.
If Sauron had known the truth he would probably have brought his full might to bare on Arragorns forces at the black gate immediately. As it was he was distracted long enough to allow Frodo to reach the top of Mount Doom.