I spent two years working on a newspaper copy desk. During that time, my coworkers and I collected examples of what we'd termed the "Department of Redundancy Department." Some reporters were notorious for saying the same thing twice, or even repeating themselves in the stories (notice the example in the previous sentence).
Now, redundancy isn't confined to repeating yourself. There are such things in writing as what I call redundant actions (or description) -- things that you don't need to say because more concise text already assumes said action is taking place.
For example: "She turned her head to look east across the desert."
This is redundant because the act of turning to look means that you're turning your head. Even deeper is the action "to look," which can stand on its own in the sentence.
Now here's the revamped version: "She looked east across the desert."
See how much stronger that sentence is now?
Redundancy is good in some areas (the space shuttle, cars, electronics, etc) but not in writing. The potential for slowing down the pace of the tale and jarring the reader from the story is too great. Mind you, redundancy is not the same thing as repeating a piece of information to draw attention to it. Especially when the original tidbit was supposed to be foreshadowing and now your MC has figured out what it means. That's just plain tying up loose ends.
The easiest way to avoid this, in my opinion, is to read your text out loud. You can hear the redundancy much more easily when speaking the words than reading them on the page. I don't know why that is, just that doing so has helped me numerous times in the past.
What examples of redundancy have you seen?