How is my "Home Base" determined, and how can I move it?

In Crusader Kings II, each noble has a "home base" of sorts: it's where their coat-of-arms appears on the map. When you're a ruler, your vassals are more likely to revolt the further away they are; it's hard to hang onto Jerusalem if you're the King of England.

I'd like to reduce this penalty for distant vassals by moving my "home base" to somewhere a little more central. (Eventually, of course, I will conquer the world and then move my capital to somewhere central like Venice. Mwahahaha.) How can I move my home base? Alternatively, how can I move a vassal's home base to somewhere closer to me (so that Jerusalem stops revolting)?

What you call “home base” is actually called capital:

As you can see, the county that's your capital is marked with a crown on the county view. And hovering over it explains what being capital means.

If you look at any other county in your demesne, the crown becomes a button you can use to move your capital:

If you press it, a dialog appears explaining that you can move your capital only once per lifetime:

NOTE: An exception to the one-move-per-lifetime rule is that a ruler can always (in peacetime) move his capital from where it currently is to the De Jure capital of his primary Title.

You can't directly move the capital of your vassals, but you can force them to do that by revoking the county that is their current capital. But you can't choose where will they move the capital to.