You can see and use contact-groups on your iOS-Device, you can even filter by them in the Contacts App, but you cannot create them or assign contacts to them.
With Contact Groups App however you can add groups, delete, duplicate and rename them, and you can add contacts to them and remove them.
You can create new contacts directly in the address book and group of your choice, update contact details, cut or copy contacts and paste them in your preferred group, even in another address book, even an Exchange-Account (see below).
Group assignments can be managed for the group (organise-button top right) or for a single contact (left-swipe).
Groups can be duplicated and then modified.
For every contact you can choose to order it by name or by company (this is not supported by Exchange-address-books).
Groups and group-memberships are stored directly in your address books (local, iCloud, Google, Yahoo, MacOS Server, …) and all your remote address books are synchronised with every connected device. No extra data is created or stored.
Exchange account notes: Exchange has categories and distribution lists or groups, but they are not shared with iOS. However, the iOS Address Book displays the Exchange folders as groups, but an Exchange contact belongs to exactly one Exchange folder, so it can not be in multiple folders at the same time (except as a copy). So the groups displayed on Exchange are actually Exchange folders when using Exchange. With the Contact Groups app, you can target new contacts to specific Exchange folders and organise them in folders by cutting or copying them from one folder and pasting them to another folder. However, under iOS, no new Exchange folders can be created, this must be done via OWA or Outlook.
Google contact groups and Yahoo contact lists
If you create groups with the Google web-app or if you create lists with the Yahoo web-app, you will not see them on your iOS-device. They are not shared with iOS.
However, with Contact Groups App you can create groups and assign contacts to them and these groups and assignments are synchronised correctly between all Apple-Devices.