The aperture is the hole that lets the light through to hit the sensor. The wider the aperture, the more light it let in, but the shallower the focus (called Depth of Field, or DOF). The smaller the aperture, the less light, but more of the image is in focus. By "more". I mean background, middle ground, and foreground, as opposed to top/middle/bottom of the actual photo.
By using a wide aperture for a shallow DOF, it is possible to have a person in focus while blurring out the background. It works best outside with lots of room, as the further away the background, the more blurred it is.
As you can see in the above images, the background is out of focus in the first, fairly out of focus in the second but keeps some detail, and in focus in the last. The apertures I used are f/1.7, f/2.0 and f/5.6. The first was taken on a bright overcast day so I was able to widen my aperture and keep my shutter speed quite fast. The field we were in was large so the trees in the background were a good 100m away. The second was taken on a sunny day, but mainly in the shade. I narrowed my aperture so I did not over-brighten the image, and to keep some of the trees behind her a little more focused. It was taken in a lightly wooded area, so the trees behind her were only maybe 5m away. The with the last one, I used a much narrower aperture to keep the detail of the archway behind her in focus. It was a very bright overcast day so I could still use a fast shutter speed. The arches were maybe 5-10m away behind her. Now, there's not that much between these apertures (it goes up to f/22!) yet you can see the difference just a few stops makes in the photos.
When changing the aperture, you must remember to change the shutter speed and/or ISO to ensure that your exposure is still correct.
It doesn't just work for portraits though; it is great for insect photography, flowers, and still life. I think that landscapes are the main subject that doesn't benefit from it because ideally you would want the whole landscape in focus. However, it might work for some abstract landscapes!