Several people have taken on the role of Captain America. Outside of Steve Rogers, the United States government gave John Walker the role, Iron Man offered it to Bucky Barnes, and Steve himself gave the shield to Sam Wilson.
Walker took over after Steve quit the role in in Mark Gruenwald and Tom Morgan's Captain America #332. Tony Stark gave it to Winter Soldier after Captain America died after the Civil War storyline. Finally, after he began rapidly growing old after losing the effects of the Super-Solider Serum, Steve handed the role to the Falcon.
This happened in Captain America Vol. 7 #25 by Rick Remender, Carlos Pacheco, and Stuart Immonen. When the world believed the Star Spangled Avenger was dead, he, in reality, went to Dimension Z and lived and fought there for 12 years. When he returned to his regular time, he faced a villain named Iron Nail, who neutralized the Super-Solider Serum in his body, causing Steve to rapidly age. Simultaneously, The Falcon risked his life to save the world. At a subsequent Avengers' meeting, Steve introduced Sam as his official replacement.
The adventures continued for Sam Wilson in All-New Captain America in 2015 and then Captain America: Sam Wilson from 2015-2017. While it was Steve who gave Sam the role and believed that he deserved it over anyone else, the two eventually had a falling out. In Captain America: Sam Wilson #2 by Nick Spencer, Daniel Acuña & VC's Joe Caramagna, Sam had a disagreement with S.H.I.E.L.D. after someone leaked confidential information about the organization remaking its own Cosmic Cube, which Sam was against. After helping take down Hydra cells, Sam quit working for the government and instead chose to work for the people.
Eventually, S.H.I.E.L.D. created the Cosmic Cube in secret, and the new Cube, Kobik, gave Steve Rogers his powers back and restored his youth. Both Rogers and Wilson then operated as Captain America. Hydra Supreme replaced Steve Rogers, and some of the world began to see Sam as a socialist version of Captain America. In Sam Wilson: Captain America #21 by Nick Spencer and Daniel Acuña, Sam officially quit his role as Captain America, reminding the people that Steve Rogers quit when he didn't believe in the direction America was headed.
This wasn't the end of Sam Wilson as Captain America, though. After working in a rescue network, helping people escape from America under Steve Rogers' fascist rule, events brought Wilson back into the fight. When the Underground convinced Sam to help them retrieve pieces of Kobik to restore Rogers back to normal, Sam saw how far the United States had fallen and took up the mantle of Captain America once again.
Sam approached Steve Rogers with the final Cosmic Cube piece. He headed it over to Steve while the Winter Soldier released the real Steve Rogers from the cube. Once Rogers returned and beat his evil self, Secret Empire ended, and the world went back to its original form. But there was still one last thing to do. In Generations: Sam Wilson Captain America & Steve Rogers Captain America #1 by Nick Spencer and Paul Renaud, Sam returned the shield with a note, saying Steve needed to inspire the people again and be the hero they need. Sam knew that the world needed Steve Rogers as Captain America, and he wanted his friend to have a chance to redeem himself.