The Eee PC (pronounced simply "ePC") is what's known as a "netbook". It's a small formfactor laptop meant for surfing the web, managing email, using Office, google apps, video conferencing. Basically everything except heavy lifting (don't do games, development, photoshop/premiere).
So it's really about personal convenience. No need to drag a full size laptop around anymore when your needs are simple. I was joking with a friend the other night that I probably have a Pavlovian reaction to opening up my Dell laptop - it saps my enthusiasm because I so associate it with work.
I think this little guy weighs about three pounds, or a little over. It has built in wireless - I turned it on, downloaded Chrome and Firefox, Google Desktop, and iTunes and I'm off to the races. I took a photo with my camera of the laptop, took the SD card out and popped it right into the SD slot built into the computer, and uploaded the photo above. It's really just very handy. Great for casually taking notes and writing, too.
I was a little worried about the small size of this netbook at first. My hands are probably larger than most, but I find the keyboard very comfortable. I touch type, and am not slowed down by the size at all. The display is 1024 pixels wide, so you don't really sacrifice much when it comes to web browsing. It is only 600 pixels in height, which makes Chrome a great browser for this device, since it has minimal clutter in the way of toolbars, etc. The computer also has a webcam built in, which I haven't used yet.
Well, I named my eee (which stands for Easy to learn, Easy to work, Easy to play) pc Cerberus - the mythical three-headed dog. I think it fits well. This is a great device - no buyers remorse at all, especially because it's so cheap compared to most laptops.