Chrome 68+: Right-click any text field and select "Emoji" or "Emoji & Symbols".

Windows: Simply press the Windows key + the period button to display Windows's emoji keyboard on Windows 10.  In case you are using an older version of Windows, you can copy-paste an emoji from emojipedia: iOS / Android / Windows.

Mac: Press CTRL + CMD + Space to display the emoji keyboard and pick an emoji.

Everywhere: Copy emojis from Emojipedia (https://emojipedia.org/) and paste it in your message

Firefox extension: Use the emoji cheat sheet https://addons.mozilla.org/en-US/firefox/addon/emoji-cheatsheet/  

Chrome extension:  You can use a third-party extension like Emoji Keyboard by EmojiOne https://chrome.google.com/webstore/detail/emoji-keyboard-by-emojion/ipdjnhgkpapgippgcgkfcbpdpcgifncb
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Custom emojis: Some device manufacturers use a set of custom emojis. See how they look here: Samsung / LG.

Android 4.4: On Android versions before 4.4, emoji emoticons might not be displayed correctly.