Notes are self signed messages that can be posted to a few relays or a lot of relays.

If you broadcast a note, you push the note from the few relays that you might have read it from to the other relays that did not already know of the note.

It's a way of saying: I like this note, I want it on more relays so it has a better chance of staying around and more people could potentially see it.

It probably helps to boost it so your followers see it too.

Here's a nice pic to illustrate: