Kodi, formerly known as XBMC, is a media player that makes streaming easier. If you've used media players on desktops and laptops in the past, think of VLC player, Windows Media Player, and other things to play videos or even stream.
What's special about Kodi? It's purpose-built for mobile devices, device sticks like the Kindle Fire, and many other things to help people play videos on any screens they have.
You've likely heard about it in regards to playing streaming services such as Netflix, Hulu, Amazon Prime, or many others. That's because you can't easily plug those websites into a computer's media player, and unlike computers with web browsers, not all TVs can open and play streaming sites easily.
That's where Kodi comes in. It's an open source project, meaning that a for-profit commercial company doesn't own it, and many projects allow multiple streaming systems to work.
Put simply, it's a homebrewed way to play what you want. If what you want isn't there, you can either ask someone to make it work, ask for ways to make it work yourself, or even start your own project.
But you're just here for the mobile app, so here's what matters.
Current Kodi App Benefits and Problems
Greater Addon Support. Kodi isn't just about Netflix, Hulu, or even YouTube. There are lots of streaming services out there, and many make easier ways to stream their content via Kodi.
Easy Local File Playback. Lots of apps can play videos. Few have the widespread name recognition and ease of use of Kodi. It's not about what works, but what's easy to use.
Support on Lots of Hardware. What if your TV doesn't have a USB or HDMI slot, but you have a video game console? If you have an Xbox, Playstation, Steam Stadia, or other devices and you're not satisfied with what they have (especially since some have agreements to avoid certain streaming platforms), Kodi works here.