With automatic syncing enabled, your contacts will be automatically synced in the background while your iPhone is idle or while you are using other apps.
If you're having trouble with auto sync, please read this article:
To enable auto sync, go to Contacts Sync > Sync, select "Auto," and choose from one of the following sync types:
- Disables automatic syncing
- "2-Way Sync"
- Sync changes in both directions, updating both your iPhone and Google
- "Gmail Contacts -> iPhone"
- Updates your iPhone contacts with the latest changes from Google
- "iPhone Contacts -> Gmail"
- Updates your Google contacts with the latest changes from iPhone
You can review past automatic syncs by going to Contacts Sync > More > Sync History.