The keywords you are looking for are 'Depth-of-field' or 'Bokeh'. You will find many answers on the site (As @ajhenderson mentions) and in short:
- At larger aperture settings (f-number is low) less of the scene is in focus. This is called shallow-depth-of-field.
- This results in blur in objects in front of and behind the subject you focus on
- As you stop down the aperture, more and more of the image is in focus and the blurring decreases.
- You need to get a lens with larger maximum aperture (say f/1.8) and a longer focal length (say 35mm or 50mm) though you can arrange things such that you can get such a blur also with typical kit lenses.
- For a normal kit-lens I would suggest setting it to 55mm (the telephoto end) and opening up the aperture to the maximum (say 5.6). Depending on the lens, you might find everything is a little soft, but you will get the feel for this.