Which one should I get?
It ultimately depends on what you are using it for.
If you are buying a device just for reading, nothing beats E-ink Kindles (Basic/7, Paperwhite, and Voyage).
If you are getting a device for light reading, with the occasional multimedia usage of apps, movies or other content, then you should get the Kindle Fire – a more versatile tablet. It is essentially similar to getting an Ipad or a Samsung Galaxy Tab; it is a tablet, albeit a very affordable one
More detailed explanation of the differences:
Why E-Ink Kindles (i.e. Basic/7, Paperwhite, Voyage) rock as dedicated e-book readers
If you have ever read for an extended period off a monitor/tablet, you will be sure to experience discomfort in your eyes or even headache due to the bright light and glare of the LCD screen.
Rather than LCD screens, E-Ink Kindles use well, E-Ink screens that are designed specifically for comfort and ease on your eyes.
With no screen glare and sharp, dark text, they are meant to replicate the feel of reading a book. In that regard, it has succeeded; reading on the E-ink screen doesn’t feel very different from reading an actual book.
Thanks largely to the energy sipping E-ink screen technology, you can easily get a battery life of an astonishing 4-8 weeks on the E-Ink Kindles.
This means that you can lose yourself in the world of your book instead of fretting over the next charge, making it a perfect travelling companion.
Weight is an important factor if you want to read comfortably for prolonged periods.
The E-Ink Kindles are lighter than most paperback books and you can comfortably hold it in one hand for long reading sessions.
Kindle is purposely designed as a dedicated e-reader. Indulge your love of reading without interruptions like email alerts or push notifications.
Why would I want to get a Kindle Fire?
APPS, APPS AND APPS
With the Kindle Fire, you are getting more than just an e-book reader, you are also getting an Android* tablet.
This means you can install android apps from the Amazon Appstore. You can even install the Google Playstore** if prefer. With the Kindle Fire, you can use your favourite apps (Facebook, Youtube, Email etc) on top of reading.
More importantly, you now have access to more sources for reading; Flipboard, Zinio, Magzter, The Economist and so on.
* The Kindle Fire runs a skinned version of Android
** Rooting and a bit of digging around the Kindle required
Henceforth this is the comparison between your normal Kindle and Kindle Fire (which is similar to tablets like iPad). Which Kindle to get ultimately depends on if you need it more for reading or multimedia consumption.