Contacts can be organized into groups on the iPhone or iPad to make them easier to find. However, it may not immediately be obvious that this is possible because Apple still doesn't allow you to create those groups on iOS 7 -- you can only view them. To create groups, you must do it in iCloud or from your computer. Here's how to create groups using both methods:
To create a group in iCloud, visit icloud.com and log in with your Apple ID. Click on contacts and then find the "+" symbol at the bottom and to the left of your contact list. You'll have an option of creating and labeling a new group. To add contacts to the group, click on "All Contacts" and drag/drop names on top of that group. Then launch the Contacts app on iPhone, and tap "Groups" at the top left side of the screen. You'll be able to view those groups by tapping them to put a check mark next to it. Uncheck groups that you'd like to hide.
Another way to create groups on the iPhone or iPad is through the Contacts app on Mac. Launch the app, then hover your mouse next to where it says "iCloud." Click on the "+" that appears to create and label a new group. Then, just like the iCloud web interface, you can click and drag/drop names onto the new group.
Unfortunately, if you're using Google for syncing contacts, the iPhone won't recognize the groups you've created there. You'll have to sync your contacts with iCloud and create groups there in order to access them.