Here's a few of my own examples of where I think the portrait orientation really works:
Similarly, portrait orientation is called that for a reason too. Most photographs you see of people are taken in portrait mode, with the head showing as well as a small amount of the neck, chest and shoulders. Sometimes, though, I think that portraits of people (and dogs) look great in landscape orientation (ie. horizontal). It can give the person somewhere to look, it can show depth, it can accentuate the features (especially the eyes if you get in closer), it can include a bit of atmosphere in the background and it can work if a bit more of the body is shown. Here's a few examples:
See also: Part 1 - Go Out in the Midday Sun
Part 2 - Ignore the Thirds Rule of Composition
Part 3 - Turn it Upside-Down!
Part 5 - Playing with the Zoom
Part 6 - Shooting Out of Focus