It is often asked how a photographer achieves the blurry background that is popular in many professional photos. The simple answer is that it's the aperture setting, but there are other factors that seem to influence the end result.
First, let's talk about aperture. Technically speaking, the aperture is an opening within the lens/camera that can be adjusted in size. The aperture setting (referred to as the f-stop) will influence how much light is allowed in for the photo. Basically, the larger the hole/aperture, the more light is allowed, which makes sense. In addition to allowing more light, the wider aperture also decreases the depth of field.
The depth of field is basically how many objects will be more or less in focus, depending on their distance from whatever you're focusing on. If you have a large depth of field (a deep focus, you might say...but I don't), more of the photo will be in focus even if the subjects are at different distances from the camera. If you have a small depth of field (shallow focus), you have just a small range of distances that will be in focus, while the rest will become blurrier as it gets farther from the target.
Here's a visual aid. Not a great one, but you'll get the correlation between the f-stop and the blurriness.
So, if you know how to change the f-stop on your camera, change it to a smaller number (which is a larger aperture--it's just the way it's written that is kind of confusing) for a more shallow focus/blurry background. I have a lens that goes down to f/1.8 (that's how you state the f-stop) and I generally keep it at that when I'm taking portraits. Many professional photographers take portraits at about f/1.2 or f/1.4, giving the scene a very dreamy look and helping the subject to stand out. I don't yet have a lens that can go that low (they're expensive).
Now, there's something I had to learn on my own, and it's that most point-and-shoot cameras a) don't allow you to set the aperture very wide; and b) even at the same f-stop, they don't seem to match the shallowness achieved by that f-stop on an SLR. I'm sure there's a technical explanation, but I only have a guess about part of it, so I won't be explaining that here.
The thing about photography is that you need to know your camera's limits and work with them. There are other factors that can increase or decrease your depth of field, or at least how it appears. You can manipulate it even if your camera won't let you get a low enough f-stop. I started practicing photography by taking lots of macro (close-up of a small subject) shots on my little point-and-shoot, and because I was zooming in, I achieved a shallower focus. Try it and see what I mean. So one way to help blurry up your background, even with limited cameras or lenses, is to zoom in on the subject.
Here's an example of what I mean. Before I got my lens that allows me to go down to f/1.8, I used my zoom lens to achieve a fuzzy background (the following two photos were taken on two different SLRs but only in that the second photo is a newer model number in the same camera series):
My focal length was 110mm on this, so the end result is a blurry background despite being at f/4.5.
Here's a comparison with a photo I took with my newer lens at f/2.2 (with the focal length being 50mm). Please never mind the tree growing from K's head.
Also, placing your subject farther from the background will obviously help, since things get blurrier the farther they are from the item on which you're focusing. Leaning someone against a brick wall isn't going to make a crazy-blurry brick wall, because it's so close behind them. But if they are standing 10 feet in front of it and you're zoomed in on their head and shoulders, it'll get blurry.
This is a concise way to explain things as I know them. I don't want to bog you down with too much info, so feel free to ask more questions in the comments.
Did this make sense or did I only confuse you more? I need feedback so I can actually be helpful!