You can use emotes for a variety of things across the discord. You can use them like you would on Twitch or via text -basically as a normal part of the conversation.
To upload custom emotes into your own server, go to your server settings and scroll down to emotes. Under the server emoji heading, you will see a brief description of how to upload emotes.
Your Nitro subscription gives you the ability to use any emote from any server you are a member of in any other server. This is known ad global discord emotes.