Aperture priority mode allows you to change the aperture (iris) of the camera, but everything else is auto adjusted for you. Use this mode if you're shooting for the shot that focuses on the subject and blurs out the background. Open up the aperture all the way (lowest number), zoom in as far as possible (helps enhance the look), and the background will automatically drop out off focus. If it's too dark (like if you're indoors), you might have to add some light to the shot so that the shutter speed doesn't automatically drop super low. If shutter speed drops too low, the shots will all of a sudden become choppy and the person's movement will look like a robot.
Great mode for beginners. This mode adjusts aperture and shutter speed for you so the exposure changes automatically. Start with this mode, but when you get to the point when you don't like the look and you want to change the brightness of the shot somehow, try manual mode so you can take more control of the camera.
Making a slow motion shot less blurry is all about shutter speed. Eric Franks breaks it down so it's easily understood. Towards the end, he shows some great sample footage that shows the difference between shooting at 1/24th of a second and 1/1000 of a second. The difference is clear.