If your iPhone or iPad displays the names of your contacts in the wrong order, for example the last name before the first name (or vice versa), you can correct it. See below how you can change them. This will change the way names appear in Mail, Messages, Phone, Contacts and other applications that comply with this setting.
First, open "Settings" on your iPhone or iPad.
In "Settings", scroll down and press "Contacts".
In the "Contacts" settings, click "Display Order".
In the "Display Order" you have two options. "First, Last" displays the first name and then the last name. "Last, First" displays the last name and then the first name. Choose what you prefer.
The "Short Name" function in the Contacts settings also determines how you view the names in applications such as Mail. With "Short Name" enabled, the application may display only one name instead of the string with the full name you just specified.
To double check, press again to return to settings of "Contacts" and then press "Short Name".
After that, press once again and then exit “Settings”.
The next time you use an application that draws information from your contacts, you will see the names in the order you just defined. If you do not see the changes yet, you may need to do so restart of the application first so that it can reload the settings. To close a running application, launch “App switcher”And drag it application up and off the screen. Tap the application icon to restart it.
Source of information: howtogeek.com