First off, to save yourself some headaches, make sure the video you're trying to transfer is in a compatible MP4 format. If your file is in a different format, like WMV, AVI, or DivX, go grab any video converter software that promises to output an iPhone-compatible file, and you should be all set.
Please note, though, that any movies or TV shows you've purchased through Apple's iTunes store use a protected video format that will not work on the Kindle Fire or any non-Apple device. The techniques described here only work on unprotected video content, such as home movies, unprotected video downloads (including video podcasts), and any unprotected digital video from other sources.
For advanced users, try using a free program like Handbrake (shown here) to convert your files using the iPad preset.
Next, connect the Kindle Fire to your computer using a Micro-USB cable (shown here on the left). Amazon doesn't include one in the box, so you may need to go out and buy one. They're easy to come by, and you may already have one for your smartphone.
Be careful not to purchase a Mini-USB cable (shown here on the right). The name may sound similar, but the connection will not work in the Kindle Fire's Micro-USB port.
Here's the part that throws most people. Logically, you would think to look in the Video menu to find your content--but Amazon reserves this space just for the purchased or rented videos you get from them.
To find your videos, go into your apps and locate the Gallery app. This is an app that is common to every Android-based device and it's there to show off your photos and videos. Open it up and your videos should be there.