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.