If you're using the official Mastodon app or some of the third party apps, the emoji are split across two menus.
You can access standard emoji by pressing the emoji button on your phone's on-screen keyboard as normal.
However, you can also access your server's custom emoji by pressing the app's emoji button when writing a post. There are no standard emoji in the app's emoji section, just the custom ones.
Each server administrator can add whatever custom emoji they want, including animations (within certain size limits). If you have some ideas for custom emoji, ask your server's admin if they could add them for you.
https://emojos.in/ uses Mastodon's API to query for the available emojos (custom emojis) of any instance.
This has can be useful in many ways: discovering quickly and easily the emojos of the instances you may have accounts, choosing an instance based on the emojos, discover new emojos to add your personal instance or request the administrators to do so, if they are open, and just out of plain curiousity/fun!