The Contacts app comes built-in to every iPhone and iPad and can't be uninstalled. Whether you want to call, iMessage, email, or FaceTime someone, the Contacts app for iOS is your central repository for staying in touch with the people that matter most to you.
The original version of the Contacts app was only available for iPad. A later version of iOS brought a dedicated Contacts app to the iPhone as well. For the most part, the Contacts app simply provides the same functionality that the Contacts tab in the Phone app does.
While both the Phone and Contacts app let you view your contacts, edit them, and sort them by groups, you still can't create and delete groups via either app on iOS. In order to do this you must either use the Address Book app for Mac or use the iCloud website, if you use the iCloud service to store your contacts, in order to create groups and organize your contacts into them. Once you've done that, iOS should automatically sync group names to both the Contacts app and Phone app. You can then show and hide groups on demand in order to find contacts faster.