What's the deal with that staff Gloucester carries around? It's not for fun; it's symbolic of his position as Lord Protector of Henry and of England.
Gloucester was given the title "Protector" when Henry was first crowned because Henry was a little baby at the time. Since babies can't rule countries very well, Gloucester's job was basically to be king without actually being called king. Now that Henry's all grown up, a lot of characters are making it known that they think Gloucester's role has become unnecessary.
Now, we don't know about you, but we can imagine that being the kind-of king for a couple of decades might be a cushy job you wouldn't exactly want to give up. We're not sure exactly how Gloucester sees thing, but we do know that he's having dreams. He tells his wife all about it:
Me thought this staff, mine office badge in court,
Was broke in twain—by whom I have forgot,
But, as I think, it was by th' Cardinal—
And on the pieces of the broken wand
Were placed the heads of Edmund Duke of
And William de la Pole, first duke of Suffolk.
This was my dream. What it doth bode God knows. (1.2.25-32)
Is it just us, or does it seem like Gloucester's dream is a warning? In fact, this dream is exactly what happens to him: he's forced to give up his position as Protector at the hands of Cardinal Beaufort and Suffolk (and Margaret). Gloucester doesn't know for sure what's going to happen, but he's uncomfortable enough to shut his wife down when she talks about the possibility of becoming queen.
Anyway, we can tell from this passage that Gloucester's staff is just as important as the title of Protector itself—in fact, the staff becomes synonymous with the position. Later, Henry tells Gloucester, "Give up thy staff. Henry will to himself / Protector be" (2.3.25-26). Did you notice how he says "staff' instead of "title"? That's how symbolic the staff has become.
It's a little weird that a fancy stick is all that distinguishes Protector Gloucester from plain old non-Protector Gloucester. At the end of the day, the differences in status between different characters in this play are kind of imaginary: it all comes down to just a stick or a shiny crown on somebody's head, and those things can be taken away pretty easily. That kind of puts things into perspective, doesn't it?