Easily send email from your WordPress site through Mailgun using the HTTP API or SMTP.

Noted That:

To install correctly this mailgun.zip.
Fisrt Download the mailgun.zip to your computer
Extract/Open mailgun.zip to Your Computer.
Then, Find readme.txt file inside mailgun.zip and Open readme.txt.
Now, Read the Requirements of this plugin. Which Wordpress Version and PHP Version are required to run this Plugin in Your Wordpress Site.
Then, Follow the the Tips Below.

Start the Tips:

1. First Download "mailgun.zip" Plugin to your Local Computer. (Click Download)

2. Then, Login to your "yourdomain.com/wp-admin" Dashboard.

3. Then, Click on "Plugins" + "Add New" from left sidemenu of Dashboard.

4. Now, Click on "Upload Plugin" button.

5. Now, Browse "mailgun.zip" Downloaded plugin from your computer, Where you downloaded mailgun.zip According to Step – 1 Above then, click on "Install Now"

6. Now, Click on "Active Plugin"

7. Then, See left sidemenu. "Mailgun for WordPress" folder is added on left sidemenu. Now, Click on "Mailgun for WordPress" folder.

Noted that: If you do not see "Mailgun for WordPress" folder on left sidemenu then, see at left sidemenu "Settings" or "Tools".

8. Now you configure yourself oR Watch video tutorial below about Mailgun for WordPress Configurtions and Settings or How to work "Mailgun for WordPress" in your WordPress site.

After Activated Plugin According to Step-6 then,

Go to "Plugins" + "Installed Plugin" from Wordpress Admin Panel Leftside Menu.
or Direct go to: https://yourdomain/wp-admin/plugins.php
Then, Find "Mailgun for WordPress" Activated Plugin from Plugin List.
Then, Click on "Settings" from Plugin that is Mailgun for WordPress
Now, Edit/Add/Config the setting and Click on "Save Changes" button,

Guide

Guide
Upload the `mailgun` folder to the `/wp-content/plugins/` directory or install directly through the plugin installer Activate the plugin through the ‘Plugins’ menu in WordPress or by using the link provided by the plugin installer Visit the settings page in the Admin at `Settings -> Mailgun` and configure the plugin with your account details Click the Test Configuration button to verify that your settings are correct. Click View Available Lists to review shortcode settings for lists in your Mailgun account that you may wish to help users subscribe to

Upload the mailgun folder to the /wp-content/plugins/ directory or install directly through the plugin installer
Activate the plugin through the ‘Plugins’ menu in WordPress or by using the link provided by the plugin installer
Visit the settings page in the Admin at Settings -> Mailgun and configure the plugin with your account details
Click the Test Configuration button to verify that your settings are correct.
Click View Available Lists to review shortcode settings for lists in your Mailgun account that you may wish to help users subscribe to

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Reffered: https://wordpress.org/