Kindle Fire is a $200 device, compared to $80 for the cheap black-and-white Kindle, or compared to $500 for the iPad.
The reason you want it more than the standard "eInk" based Kindle is that it can play videos, run apps, and show books in color (like comic books). The downside is that it weighs twice as much as the standard Kindle, lasts only 10 hours on battery (vs 30 days for the standard Kindle), and only works with WiFi (no mobile phone connection).
It's a 7 inch screen (14.6 ounces0 compared to the iPad's 10 inch screen (21.3 ounces). This makes the Fire better for traveling than the iPad. But, since I travel with my iPhone and MacBook Air, I really don't need another device to watch video. What I need is something that won't run out of batteries. For that reason, I'm going to travel with the eInk Kindle Touch (7 ounces), not the Kindle Fire. (In addition, the Kindle Touch as a mobile phone connection that can be used in emergencies when no WiFi is available, unlike the Fire).
There are plenty of annoyances with device. Some content ("The Watchman" and "The Economist") is shrunk and can't easily be expanded (The Watchman not at all, The Economist with zoom-and-pan for every page). The same thing happens to web-pages: the iPad and standard Kindle have been around long enough for people to format content for them, but they treat the Kindle Fire as a full-sized desktop screen, and not the 7 inch screen that it really has.
The web-browsing goes through Amazon's cloud for "acceleration". Instead of browsing the web directly, you go through Amazon's servers, which strip out all the stuff that slow down web browsing. In practice, I don't think this works so well. Yes, some web pages are "snappier", but at the same time, some things behave oddly. I'd have to spend more time at it, but I think my experience on the iPad is better.
But for all such annoyances, it works much like the iPad. As his biography points out, Steve Jobs was really angry at how much Android (which runs the Kindle) copied most of what makes the iPhone/iPad cool. It even does a couple things better, such as cloud integration. Your books, magazines, music, videos, and apps have two selections, those in the Cloud, and those on the Device (demonstrated in the picture above), and downloading from one to the other is simply the touch of a button.
Here's the verdict: it's not as polished as the iPad. Your 2-year-old or grandparent can't pick it up and immediately start using it, like she can with the iPad. But that probably doesn't matter, since 99% of the time is spent with the content (reading, watching, playing) rather than with the device software. You have to ask yourself if an extra $300 is worth the difference in usability/polish for the small amount of time between watching videos, reading books, or playing games. It probably is important for the very young or very old, or the very geeky, but probably not so important for everyone else.
Sure, the recipient is going to be disappointed you didn't pony up the extra bucks for an iPad, but otherwise, the Kindle Fire is going to make a great gift for Xmas. I think they will spend more time using it in the coming year than almost any other present you could give them.
I'm not part of Amazon's affiliate program. Following the links on this page to Amazon's site gives no benefit to me (which you can verify with View Source and check out the links). I thought I'd point this out since Amazon's evil affiliate program leads to so much spam. These are my honest opinions.
Here is a Kindle Fire review from The Daily. It agrees with the points I made. But whereas it stresses "it doesn't do anything exceptional well", the real point to remember is that it does everything adequately well. And, that's only during the 1% of the time you aren't watching a movie or reading a book, at which point, the experience is roughly the same.