We know that a rose by any other name would smell as sweet, so why are the characters in this play fighting about roses so much?
Well, when York tells us he's going to "raise aloft the milk-white rose" (1.1.266), he's not actually talking about flowers; he's talking about the House of York. The Wars of the Roses, a long series of battles between the houses of Lancaster and York, are the backdrop for the Henry VI trilogy. These people didn't fight with roses (if only); the images on their coats of arms were roses. The Lancasters had a red rose, while the Yorks had a white rose.
That's how the war got its name.
The main argument between the Lancasters and Yorks was over who should be king and who had the right to rule. Countless people died while the crown was handed back and forth between York and Lancaster men. It was a tough time for England.
But why write about the Wars of the Roses at all? What's the big deal?
Well, Shakespeare's audiences couldn't read about historical figures in the library or online—they went to the theater to learn about kings and queens of years past. So, in one sense, Shakespeare is giving his audience a history lesson.
On the other hand, Queen Elizabeth I had passed a law that no one could write about current political affairs, so Shakespeare used the past to comment on the present. He wrote about old historical figures and situations that were similar to ones happening around him. That way, he could comment on current events without actually seeming to; he could always just say, "Hey, this is history. I'm not doing anything wrong."