Each Mastodon server has its own set of custom emoji. These can be any image :cate: or short animation :rickroll:

To access your server's custom emoji in the apps, tap the 😃 smiley icon within the app itself when you're writing a post (and NOT the smiley on your keyboard, that's just for normal emoji).

To access custom emoji on your server's website, click the emoji picker icon 😂 in the post writing box.

If you want more custom emoji added to your server, contact your server admin.