install

Short version

1) Delete the old folders first. Never override.
2) Download the AddOn.
3) Right-click and extract it.
4) Then move the folder(s) to:
• Retail: World of Warcraft / _retail_ / Interface / AddOns
• Classic: World of Warcraft / _classic_era_ / Interface / AddOns
• Cata: World of Warcraft / _classic_ / Interface / AddOns
• PTR: World of Warcraft / _ptr_ / Interface / AddOns

Full version

Step 1) Delete your old AddOn files (this will NOT touch your settings, because they are stored in the WTF folder)
Note: Never overwrite/replace AddOn files. Always delete the old folders first.

Step 2) Download your AddOn from the website

Step 3) Rightclick the .zip file and select "Extract All..." (Windows)

Step 4) Open the extracted folder and check the capitalization
Example: For ElvUI the following 3 folders (case matters): ElvUI | ElvUI_Options | ElvUI_Libraries

Step 5) Move them to the correct WoW folder: Either _retail_ or _classic_ or _classic_era_ -> Interface -> AddOns
Use the method shown below to find the wow location by using the Battle.net Application. If the AddOns folder is not there, go ahead and create one.

Many people have multiple World of Warcraft folders on their system and don't know about it.
Here is a quick screenshot guide how to check for the correct folder path.

Open your Battle.net app and select World of Warcraft - then click on the "Options" dropdown (Cogwheel icon)

Click on "Show in Explorer". This will open the correct folder for your WoW as you can see here: