When Aragorn and Co. arrive at the Black Gates of Mordor on Mission: Distract Sauron, Sauron decides to have a little fun at their expense. He wants to toy with them like a cat plays with a mouse. So rather than attacking and killing them all at once, he sends out a group of representatives from behind the Morannon (the Black Gate). The leader is a huge man, dressed all in black, with a black helmet. He rides a hideous thing that looks like a horse, but with a skull-like head and red fire in its eye sockets and nostrils. Lovely, no? This guy's a real peach.
This man is so far gone in his service for Sauron that he has actually forgotten his own name. He has literally no identity beyond his job as spokesman for Sauron. So his "name" is now the Mouth of Sauron. He is also the Lieutenant of the Tower of Barad-dûr. He is one of the men of Númenor who began to worship Sauron, "being enamoured of evil knowledge" (5.10.30). Such men are called the Black Númenoreans; Faramir mentions them in passing ("Many [Men of Númenor] became enamoured of the Darkness and the black arts" [The Two Towers 4.5.117]).
The Mouth of Sauron mocks Aragorn and Gandalf at every opportunity. He also proudly produces Frodo's gear—the mithril shirt, the Lothlórien cloak, and Sam's short sword—to frighten and threaten Gandalf by implying that the hobbits have been killed. Like all of Sauron's other servants, the Mouth of Sauron relies on fear and intimidation to keep other people in line.
But also like many of Sauron's servants, the Mouth of Sauron uses fear on other people because he is really cowardly himself. When Aragorn catches his eye and stares him down, the Mouth of Sauron staggers back as though he has been physically hit. And when he sees Gandalf looking angry, the Mouth of Sauron jumps on his evil horse-thing and rides away in fear. He talks a good game, and he enjoys other people's pain, but he is nothing without the strength of Sauron behind him.
As far as the plot goes, the Mouth of Sauron's main job is to show both the good guys and the reader that the Enemy has stolen Frodo's possessions and connected them to Gandalf and Aragorn. When last we saw Frodo at the end of The Two Towers Book 4, Chapter 10, he was being taken captive by orcs. Is he even still alive? What has happened to him? By bringing out Frodo's stuff, the Mouth of Sauron reminds the reader of Frodo's danger at the end of The Two Towers and raises our suspense about Frodo's fate before we move to Book Six and the Mordor section of The Return of the King.