Create groups on iOS devices with iCloud
Some groups are created by default, such as the All Contacts group, which can't be deleted. Other groups could include Gmail contacts (created when you add your Gmail account in the Mail app) and Facebook.
Unfortunately, and this is probably the reason you're reading this, Apple doesn't provide a way to create new groups within iOS, so you can't do it directly on your iPhone or iPad.
The answer - one method, at least - is to use iCloud.
How to use iCloud to create contact groups
Step 1. Log into www.icloud.com using your Apple ID.
Step 2. Click on Contacts and then click the groups ribbon (the red icon with two people) which is on the left-hand page when viewing All Contacts.
Step 3. The left page changes to a list of Groups (only those groups stored in iCloud are shown). Click the + button at the bottom to add a new group.
Step 4. Type a name for the new group and press Enter to save it. To change it after this, double-click its entry in the groups list.
Step 5. To add contacts to the new group, click on the All Contacts group and locate the first person to be added (you can use the search bar to find them quickly).
Step 6. Drag their name on top of the new group and drop it to add it to that group.
Step 7. To add more contacts, repeat steps 5 and 6, but you can add multiple contacts at once by pressing Ctrl (on a PC) or Command (on a Mac) and clicking on each contact in the list that you wish to select. Then, drag one of the highlighted names to the new group and they will all be added.
It's possible to add names to more than one group, and you can create as many groups as you like.
Step 8. Launch the Contacts app on your device (iPhone, iPod touch or iPad) and you should see the new group appear almost immediately - as long as you have an internet connection.
Until Apple builds in a function to create groups directly within the Contacts app, this is the best way to do it.