Twitter added support for GIFs way back in 2014, but you had to find them on a separate site and upload them yourself.
Now, you can just click the GIF button when you compose a tweet or message and browse popular categories or search with a few keywords to find a GIF. You'll only be able to post one GIF at a time and you can't share a regular, not-moving photo and a GIF in the same tweet.
The GIF search is available on both Twitter's website and the Twitter apps for Android and iOS. If you don't see the option right now, don't panic; it's rolling out to everyone in the next few weeks.
Massive GIF search engine Giphy and Riffsy, makers of the GIF Keyboard, are supplying the GIFs, so you'll never have a hard time finding the right image for what you want to say.