install

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 "ElvUI" & "ElvUI_OptionsUI" (cAsE mAtTeRs) (without the quotes)
Step 5) Move them to the correct WoW folder: Either retail or classic -> Interface -> AddOns

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 highlight the correct folder for your WoW version ( retail or classic ) as you can see here:

Short version

1) Download the AddOn.
2) Right-click and extract it.
3) Then move the folder(s) to:
• Retail: World of Warcraft / _retail_ / Interface / AddOns
• Classic: World of Warcraft / _classic_era_ / Interface / AddOns
• TBC: World of Warcraft / _classic_ / Interface / AddOns
• PTR: World of Warcraft / _ptr_ / Interface / AddOns