Using Core Data, iCloud and CloudKit for syncing and backup and how it works together

It's like this:

• Core Data on its own, is completely local and does not automatically work with any of Apple's cloud services.
• Core Data with iCloud enabled turns on syncing via iCloud. Any changes you save in Core Data are propagated to the cloud, and any changes made in the cloud are automatically downloaded. The data is stored both in iCloud and in a local persistent store file, so it's available even when the device is offline. You don't have to write any cloud-specific code, you just need to add listening for incoming changes (which is a lot like changes made on a different managed object context).
• CloudKit is not related to Core Data. It's not a sync system, it's a transfer system. Meaning that any time you want to read/write cloud data, you need to make explicit CloudKit API calls to do so. Nothing happens automatically. CloudKit does not store data on the device, so the data is not available if the device is offline. CloudKit also adds some features not available to Core Data with iCloud-- like public shared data and the ability to download only part of the data set instead of the whole thing.

If you wanted to use CloudKit with Core Data, you'd have to write your own custom code to translate between managed objects and CloudKit records. It's not impossible, but it's more code to write. It might be more reliable but it's too soon to say for sure.

I wrote a blog post describing CloudKit from the perspective of someone who's used Core Data and iCloud in the past.

Update, June 2016: As of the most recent documentation for NSPersistentStoreCoordinator, everything related to Core Data with iCloud is marked as deprecated. As a result it should probably be avoided for new development.

With iOS 13, Apple announced new features in Core Data to better work with CloudKit. The main addition is NSPersistentCloudKitContainer which basically manages syncing between Core Data and CloudKit for you.

You can learn more in the WWDC session Using Core Data with CloudKit.

Apple also released a nice collection of docs for this very usage: Mirroring a Core Data store with CloudKit.