I'm a completely self-taught photographer. By that I mean, I have never taken a photography class or gone to school for photography. Everything I have learned has come from books, other photographers and practice. Because of that my explanation of things may not be perfect, but it's photography as I understand it.
Photography is all about letting the light in. In fact, I know from homeschooling my children that photos is the Greek word for light and graph the Greek work for to draw or write. So photography is "drawing with light". Aperture is only one piece of the puzzle, but a very important one. Aperture controls depth of field. Here is a great, more technical, explanation than I am about to give. Put simply, a "narrow depth of field" gives you the "blurry background" that so many people love.
Enough technical stuff, you can research it on your own if you like. Here are some examples of photos I took yesterday. I put my camera in aperture priority, set the aperture, and let the camera do the rest. Here are the results:


Notice the difference in the background of these photos? Now, I don't normally shoot in aperture priority. I shoot in full manuel, but aperture priority is a good start for anyone looking to improve their photo skills.
Obviously, these are not portraits, but it gives you a good real-life example of how aperture affects depth of field.
So, if the background looks so great and "blurry" at 1.8, why not shoot at 1.8 all the time? Because when you use a narrow depth of field, you also have a much smaller area that is in focus.

If you look at this photo, Tucker's eyes are tack sharp, but out of focus at his bangs, nose, mouth, etc. I happen to love this type of photo. Its my favorite look for a close up portrait. It doesn't work so well for a photo with two or more people, however. It also takes some practice to be able to nail focus at this aperture. While teaching Tilly, I had her begin at and aperture of 3.2 so she could learn to nail focus there before shooting more wide open.
And now to answer the question that inspired this post. I had a reader ask, "what I did" to get the cool effect in this photo of Mordecai:

The only thing I "did" was shoot at an aperture of 2.2. The "effect" you see is simply the light shining through the leaves and trees. This is what we call "bokeh".
I did of course post-process this photo, as I do to most of my photos.
Here's the SOOC (straight out of camera) version. Well, not quite straight out of camera. I converted it to a jpeg (I shoot in RAW) and resized for the web, but that's it.

As you can see, the bokeh is there in the SOOC image. I simply cropped, brought out the color already presen,t and did the adjustments I normally make to my photos.
One of the big advantages to custom photography is just this. Each image is adjusted to look make it the very best photo possible. This is exactly why I sell choose prodominately prints and not digital files. There is so much art and creativity that goes into making a portrait. Post processing truly is an art form in and of itself.
And because I know you're going to ask, here are some of my favorite photography resources:
I process my photos with Photoshop CS3. Nearly everything I've learned about post processing has come from this book of Scott Kelby's.
I get asked constantly to recommend books for beginner photographers. My first bit of advice is read your camera manuel. Yes it's boring. Read it anyway. Mine is falling apart it's been read so much.
I love the book National Geographic Photography Guide for Kids. Unfortunately it's now out of print, so it costs a fortune! I like it because since it's written for kids, it's easy to understand! If I find a new favorite, I'll let you know.
Happy shooting!