Emotes are truly the bread and butter of twitch chat, and they are one of the main reasons why the website isn't your ordinary live streaming platform. In order to get started with understanding emotes you will most likely have to download some sort of extension in order to have many of them even appear on your twitch chat. This is optional but strongly recommended as you will only see certain emotes that twitch allows and the rest will appear in word form (BTTV recommended).
Emotes are typically artwork or photographs shrunken down to the size of regular emojis and usually have a deeper meaning or meme origin. Every emote can be typed as words as well as emote form, and a good example of this is the emote Kappa. To use it, you can click on the emote for Kappa in the emote directory on the bottom right side of the chat shown as a smiley face button or just type the keyword "Kappa" in the text box, hit enter and it will appear as the emote in the chat.
(Variation of Pogchamp)
Any emotes you see that you don't know, you can hover your mouse over top of it to see the name. In terms of when and how to use these emotes, each one has it's own unique meaning and are meant to be used in certain contexts. In the case of the example earlier, Kappa, It is meant to be used in situations of sarcasm, for example, if a streamer says something like, "I am the best in the world", the chat might react by typing a lot of Kappa's as if to say "Ya right".
Kappa is just one of the many "Global" emotes that the average twitch user has memorized and use daily but as shown with the two above, there are also different types of emotes that are just as prevalent and important to know. (More detail below) It make take a while to get a good handle on all the emotes, especially since more are constantly being adding and each channels can be different, but after some time and experience it will come naturally.
This is by far the simplest type of emotes you can use on twitch as they are available for everyone and are the standard keyboard emotes you may have used while texting or talking over the internet. Even though they are some of the simplest emotes, they are still used a uprising amount, and especially ":)" AKA "smile", and "D:", usually just referred to as "capital D colon". Some more examples are ":p", ":D", etc. For more on why these simple emotes are still relevant in twitch chat click the button at the bottom of the page to go to Memes and Meta page.
Global Emotes are typically the foundation of basic communication in Twitch chat, they are available to everyone and are some of the most commonly used emotes in chats across the platform. Knowing the meanings behind these are a great start to being able to use Twitch chat to it's fullest potential, and it can be especially useful for getting to know subscriber emotes, as many of those are variations of "classic" global emotes. Global emotes also have their own meta to an extent as some are used much more frequently than others, for example "LUL" being a very prevalent emote (top right of image), mainly based on it's utility and flexibility to be used in many contexts. For more information on the meta/usage of Global emotes click the button on the bottom of the page.
Each different type of emotes are essential to the foundation of Twitch chat but out of all the types, subscriber emotes are what make each individual chat and community unique and provide great opportunities for creative emotes and memes. This is truly where the emote "meta" comes from as each streamer will has their own certain number of subscriber emote slots that they have the option to fill up with existing emotes or create new ones, sometimes by themselves or by a designer. By subscribing as a viewer for the 5$/month you will unlock all the subscriber emotes and be able to use them in other streamers chat as well. If a lock symbol appears on them like shown here, then you have not subscribed and unlocked them yet.
Memes and Meta
The last type of emotes, and probably the least common, are Turbo emotes. These are emotes unlocked when you subscribe for $8.99 to the monthly Twitch subscription program called Turbo (different than subscribing to streamers). This subscription unlocks multiple things including ad-free viewing, a chat badge, custom chat username color, extended broadcast storage and the ability to unlock new sets of emotes, either glitch or monkey. (Glitch is currently chosen currently here)
A useful tip, if you only know part of the word for the emote and if you can't find it in the directory, you can do a faster search for it by simply adding a colon before the first few letters and it will give options to choose from like a search engine. Ex. :Kappa
For an in detail emote directory of every emote on Twitch, Check out: https://www.frankerfacez.com/emoticons/?q=Pog&sort=created-desc