This guide is slightly deprecated as V3 of 7TV's browser add-on now supports 7TV, BetterTTV, and FrankerFaceZ emotes out of the box, and has some great quality of life chat features that are currently not found in any other emote add-on. You should consider going to 7TV's website and downloading their browser add-on from there.
7TV, BetterTTV, and FrankerFaceZ, more commonly stylized as 7TV, BTTV, and FFZ, are 3rd party emote services for Twitch chat, which aim to enhance a viewer's chatting experience on Twitch by allowing for more emotes to be used in chat, free of charge. Think of it like Discord emojis.
If you're wondering what those weird phrases people in chat seem to be spamming are, it's probably an emote from one of the above mentioned 3rd party emote services. Having the ability to see those emotes is pretty essential, considering the popularity of those 3 emote services, especially in larger channels.
Step 1: Install the FrankerFaceZ add-on for your browser of choice.
Go to frankerfacez.com and click on the download button.
If you're on a Chromium based browser (Brave, Chrome, Edge, Opera) it'll redirect you to the Chrome Web Store where you can install it from there. Otherwise, if you're on Firefox or another browser, it'll install through the website.
The add-on is available as a user script as well.
Step 2: Once installed, go to twitch.tv and click on the FrankerFaceZ button in the top right. A menu should then pop up.
It should be next to the get bits button or your profile picture.
Step 3: Click on the add-ons button.
Step 4: Enable the 7TV Emotes and BetterTTV Emotes add-ons.
You're now done. Enjoy your less scuffed Twitch chat experience. 
You can also install a program called Chatterino on to your computer and keep chat tabs open on your favorite channels or offline chats.
The version linked here is regular upstream Chatterino 2 (which now supports 7TV emotes out of the box), but other forks of Chatterino exist as well, such as 7TV's fork of Chatterino, Chatterino7, which has additonal 7TV related features that are not currently present in Chatterino 2.
Chatterino Homies and Dankerino are also other noteworthy forks.
And for mobile support, as the official Twitch app doesn't natively support 7TV, BTTV, and FFZ emotes, install Chatsen or Frosty, which are both open-source chat apps available on iOS or Android.
Or alternatively, DankChat, available on Android only.
StreamElements' chat overlay natively supports 7TV emotes, while Streamlabs' doesn't. If you're using Streamlabs' chat overlay, you can instead use jChat or ChatIS, a fork of jChat with additonal features.
(But you should consider just switching to jChat or ChatIS regardless if you're using one or the other.)
If though for some reason you wish to continue using Streamlabs' chat overlay and you're also using Streamlabs OBS (why), you can refer to the image below on how to enable 7TV emotes on Streamlabs OBS:
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