Beregond is a guardsman of Minas Tirith and a member of the Third Company of the Citadel. He first approaches Pippin with orders to teach Pippin the passwords of the city and start answering some of his questions, now that Pippin has sworn an oath to Denethor.
But Beregond frankly admits that one reason he wants this duty is to find out more about Pippin, since he has never seen a "halfling" (5.1.91) before. Pippin tries his best to dodge the questions about his travels—though he does let Aragorn's name drop, which Gandalf told him he shouldn't. But he and Beregond bond over horses and Pippin's appetite for food, making Beregond seem like a nice, upstanding guy.
Beregond's main purpose in the novel is to be our touchstone in Minas Tirith. He introduces the layout of the city of Minas Tirith and the surrounding countryside. He also confirms our view that Faramir is Gondor's next best hope (after Aragorn, of course). In nearly all of the conversations Pippin and Beregond share before Faramir returns to Minas Tirith, Beregond repeats some variation of, "I wish Lord Faramir would return. He would not be dismayed" (5.4.26).
But more importantly, this guy also gives us an ordinary person's perspective on the threat of Mordor to Gondor. He explains to Pippin that the people of Gondor have all grown up in sight of that shadow. They can literally see Mordor from Minas Tirith. He sees that "[the shadow] is growing and darkening now; and therefore our fear and disquiet grow too." (5.1.135).
Beregond's view of Mordor as a constant, terrifying reality that has been growing worse and worse over the past year makes the power of Mordor over Middle-earth seem less abstract and more immediate (and threatening) to us. Sauron's war does not only affect the characters we know and love. It also influences the daily lives of every single man, woman, and child in Minas Tirith—the average Joes of Middle-earth.
When Pippin comes to tell Beregond that Denethor has gone off his rocker, Beregond worries a bit about the rightness of leaving his post to help Faramir. It's his duty to stay put and guard the city. But he knows that his true moral duty is to go and save Faramir from his insane father, even if he has to desert his post to do so.
When Pippin and Gandalf arrive at the House of the Stewards on Rath Dínen, they find Beregond defending an unconscious Faramir with his drawn sword against torch-wielding servants of Denethor's house. It's quite the scene. These servants all yell that Beregond is betraying his oath as a Guard of the City by refusing to obey Denethor's orders. But isn't Beregond actually obeying a higher moral law by not, you know, burning people alive? We think so. But the guards helping Denethor gather firewood are too busy being sheep to think much about the consequences of their actions.
Still, in the aftermath of this whole battle to save Faramir from his own dad, Beregond does get into trouble. Because Beregond has broken the official rules of the Guard by leaving his post against orders (and killing three other Guardsmen), he loses his position as Guard of the Citadel.
But because Aragorn isn't an idiot, and understands Beregond's good intentions, he doesn't give Beregond the death penalty for desertion (like he's supposed to). Instead, Aragorn appoints Beregond to the Guard of Faramir (the White Company), since Faramir is the newly named Prince of Ithilien. So now, Beregond is part of the White Company, and he gets to focus on protecting Faramir full-time. It's a win-win situation.