Fun fact: Though we don't see it yet in this play, there's about to be a civil war in England. And guess what it's called: The Wars of the Roses. So yeah, the red and white roses mean a whole heckofa lot in this play.
In Henry VI, Part 1 we witness a scene in Act 2, Scene 4 in which Somerset and Richard ask their followers to choose either red or white roses to indicate their allegiance. White roses represent the house of York (a.k.a. Team Richard) and red represent the house of Lancaster (a.k.a. Team Somerset). Team Richard wins pretty handily, to Richard's delight. So here, then, we can see the roses not only representing the two houses, but also popularity, with Richard clearly preferred over Somerset.
Later, however—in Act 4, Scene 1—despite a pretty decent attempt to quell the arguing between Richard and Somerset's factions, Henry VI makes what we'll call a rookie mistake: He pins a red rose on, while claiming to like both men equally. He may mean it—he generally seems pretty earnest in his communications—but actions speak louder than words sometimes, and this is totally one such instance. Henry could say all day long that he loves them both, but so long as he has Somerset's red rose pinned to his chest, no one is going to buy it.
At this point, the roses also become symbols for Henry's naiveté. He goes from giving a pretty good pep talk about making peace to completely undermining himself, just like that, when he decides to sport the red rose. So the roses not only represent this unrest between the houses of York and Lancaster, but also the inability of the King to foster stability. Which, of course, only becomes more true when the Wars of the Roses breaks out in Henry VI, Part 2.