Your global Discord username is visible everywhere you use Discord. A server nickname is how you appear only inside one community — useful when you want a readable practice name without renaming your whole account.
On desktop, right-click your name in the member list and choose Edit Server Profile or Change Nickname, or open the server menu and edit your profile from there. On mobile, tap the server name, open your profile, and edit the server profile fields.
Pick something people can recognise in voice and text. Do not impersonate staff or other members, and keep nicknames within server rules — no NSFW or provocative text. Your global username is still changed under User Settings → My Account if you need that instead.