Each Mastodon instance has its own optional custom emoji. They work across the Fediverse, and you'll sometimes see them on other people's posts.
Custom emoji appear in your instance's emoji picker () just above the standard emoji.
If you have good ideas for custom emoji, try asking your instance admin if they could add them
(You can find your instance admin by clicking on "About this server" at the bottom left of the screen)
If your Mastodon instance admin is unsure about how to create custom emoji, there's a guide by @laura at: