This piece appears to use the method known as "short rows". It involves knitting to a certain part of the piece (not the end of the row), slipping a stitch, wrapping it, then slipping the stitch back onto the left needle, turning and knitting back the other way.

I probably made that sound more difficult than it really is. Run a search for short rows and I'm sure you will find it explained much better than I can do.

It appears to me that the pillow is knitted with a technique called intarsia See link below.

Here is a link that might be useful: Intarsia

I am familiar with intarsia. This pillow looks like there is some kind of shaping involved. Intarsia has zigzagged line. This looks like some lines are curved and rounded. I don't know how to do this.

Many people will think I am mistaken in assuming that this piece uses the technique of short rows, since they think of it as a technique for sock heels or other shaping. But I really think short rows is the correct answer here. I am supplying a link to a DIY episode in which short rows were used to make a backgammon board. It is a totally flat piece, but still uses this technique.

Here is a link that might be useful: short rows

I think you are on to something with the short row technique. Now I have to find a tutorial where I could learn to do it. Do you know of any good knitting books where this is explained?

I am out of town this weekend and do not have access to my books, but I found a couple of videos on YouTube, by Cat Bordhi.

Part 1:
http://www.youtube.com/watch?v=_yVikAvPuE4

and

Here is a link that might be useful: part 2

I have made a cape and a ruffled scarf totally using short rows. It is not difficult, just takes a little practice, and is a good technique to know as many patterns call for short rows. Just google 'knitting short rows' and you will find a lot of good instruction sites.

Here is a free intarsia pillow pattern. It is not the same design as what you are looking for however, it give you an idea of how it is knitted.
I am interested in seeing some short row patterns that would be similar to this. Please post links if you can find them.
Thanks

Here is a link that might be useful: Intarsia Pillow