You know the drill: when you see a Marvel movie, you keep your butt in your seat until the bitter end, lest you miss the now-standard end-credit scenes. Sometimes the postcredits scenes end up meaning nothing and are just fun interludes, where others connect the Avengers in the Marvel Cinematic Universe. Black Panther's two postcredits scenes are both important to the MCU at large, but the second one is significant in the sense that it connects back to a previous Avengers movie. (Warning: spoilers follow!)
That previous point is important, because unlike many of the MCU movies, Black Panther only features a couple characters (besides T'Challa himself) that we've already met in an Avengers movie. That lets the movie stand on its own, but the end credits scenes — especially the final one — remind us that Black Panther, like Wakanda, is not alone in its world. So let's get back to the scene: in it, we're in a village in Wakanda. A couple of kids stare down a person sleeping, and when he stumbles out into the sun, we see it's Bucky Barnes (Sebastian Stan). He's greeted by Shuri (Letitia Wright), who's clearly there to check on him. She asks how he is, and says he still has work to do. Confused? Don't remember when you last saw Bucky? Here's what you forgot.
The last Avengers movie Bucky Barnes was in prior to this was Captain America: Civil War. After being on the run — as the Winter Soldier, he had committed ghastly crimes — Cap has to keep him away from Tony Stark's wrath, whose parents Bucky had killed in 1991 while he was brainwashed by Hydra. Stark strips Bucky of his robotic arm, and then Cap smuggles Bucky to Wakanda, where he has asylum. He chooses to go back into cryogenic sleep (gotta keep looking young) since he's unable to fight the brainwashing himself. (It's also interesting that Bucky ends up in Wakanda since T'Challa originally believes Bucky to be responsible for his father's death, as he'd been framed by Helmut Zemo.)
So to connect the dots, Bucky has been in recovery, but is now out of cryogenic sleep, and being cared for by the brilliant Shuri — which connects back to an earlier easter egg in the movie, when she receives a dying Everett Ross and remarks that he's "another broken white boy to fix." Those words confirm that she either has fixed Bucky or is well on her way. Arm or no arm, he certainly looks great when he greets her, which bodes well for his future in May's Avengers: Infinity War.