When photographers talk about the rule of thirds, they say sometimes they will break the rule and still the photo will have great composition. When should I stick with the rule of thirds and when should I break it? And when I break it, what should I use instead to drive eyes to specific points in the photo?
The rule of thirds to me is a rule of thumb, a reminder not to mindless frame my subject dead centre of the frame, or else I will probably end up with static or boring images overall.
As a beginner, it's a good rule to keep in mind. Not to blindly follow, but to help encourage you to try different framing, perspectives and so forth. As an experienced photographer, you'd probably not even think about it but you'd naturally tend to frame subjects off centre to make them more interesting.
Specific situations where rule of thirds might be "broken"? I would say primarily this is where symmetry is the focus of the image:
if you have a nice, symmetric water reflection, you might place the horizon in the centre of the image to give equal space to the subject and its reflection
in landscapes, if you have an interesting sky you tend to place the horizon towards the bottom of the image, or if sky is bland, place the horizon towards the top. But if you have an interesting foreground and a dramatic sky, you might give them equal weight
close ups of people and pets, like the dog in mattdm's link, especially where there is nothing in the background to balance off the subject. If the subject is interesting and engaging enough, centre placement might be all that's needed.
symmetrical subjects, for example the Taj Mahal. Beautiful symmetry might be accentuated by centering it in the image.
portraits, especially formal ones tend to be centred. Environmental or street photography would be different, where the background can be very important to the image so it needs more weight than a plain paper/muslin background.