Online backups are a useful component of a well-balanced backup strategy. Whether you rely primarily on cloud storage for backups (see "Backup Basics") or use the cloud to supplement local backups such as bootable duplicates (see "Bulletproof Backups"), it's crucial to understand how you will go about restoring your data after disaster strikes.
Disaster is the operative word here. If you merely need to restore a few individual files or folders, usually that's simple enough--typically you use either the backup client software installed on your Mac or the backup provider's website to specify which versions of which files you want, click a button or two, and wait for the files to download. No big deal.
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But what if your entire hard disk dies and needs replacing, or your Mac is stolen and you have to start over with a new one? Such situations require a different strategy, because your online backups almost certainly don't include every single file on your Mac; and in any case, even with a fast broadband connection, you may be looking at days or weeks to restore whatever data you keep in the cloud.
One way or another, you must first get your Mac back to a state of basic functionality, and then--perhaps by stages--restore your crucial missing files from the cloud. How you go about that depends on what other backups (if any) you have available.
If you have no other backups
Let's start with the least pleasant scenario: Your only backups are in the cloud, and you have no local copies of your data at all. You have to do more work and wait longer to get up and running; but if you backed up all your crucial files, you will return to a happy place in due time.
Set up OS X: Your first step is to make sure that your drive has OS X installed. New Macs, of course, come with OS X already installed. If you've had to replace a defective drive with a new, empty drive, you'll need to install OS X on it before doing anything else.
If your Mac shipped with an older version of OS X that included physical installation media (a DVD, CD, or flash drive)--or if you planned ahead and made yourself a recovery volume using the OS X Recovery Disk Assistant--then just start from that media and run the installer. Newer Macs (those released in the past two years or so) have no separate OS X media in the box; instead, they rely on OS X Internet Recovery. Hold down Command-R as you restart the machine, and follow the prompts to redownload Lion or Mountain Lion from the Mac App Store and install it on the new disk.
Get your backup or sync software up and running: After getting your Mac working again, your next step is to download and install whatever cloud backup or sync software you used. Run the software and sign in with the same account you used previously.
What happens next depends on the type of software you used:
For backup software, such as Backblaze, CrashPlan, or MozyHome, follow the instructions for restoring the latest copies of your backed-up files. (You might want to skip restoring backups of email, contacts, and calendars, as I'll discuss in a moment.) Restoration speed depends on the throughput of your broadband connection. If you find that it's too slow for your needs, you can either try moving your Mac to a location with a faster connection or request that the cloud provider ship your data overnight on a hard drive, DVD, or flash drive (for an extra fee, naturally).
While you're waiting for your files to download or arrive at your door, you can work on several additional restoration steps.
Reinstall your applications: Most cloud backup services don't back up your apps. You'll have to reinstall them from the Mac App Store (Apple menu > App Store), download them from the developers' sites, or use original installation media to get all your apps back.
Redownload purchased media: Using iTunes, you can redownload previous purchases of media such as music, movies, TV shows, books, and iOS apps (which you may not have included in your online backups or syncs). And if you signed up for Apple's $25-per-year iTunes Match service, you can download fresh copies of all your music tracks (even those not purchased from Apple).
Use Photo Stream to restore photos: If you previously enabled iCloud's Photo Stream feature, you can open Aperture or iPhoto , make sure it's still enabled (check the Photo Stream preference pane in either app), and sit back while up to 1000 of your most recent photos download to your Mac.
Sync email, contacts, and calendars: If you rely on cloud-based services for email, contacts, and calendars--particularly iCloud, Google, Exchange servers, and (for email only) other IMAP servers--getting your data back into apps like Mail, Contacts, and Calendar is usually as easy as signing in to your account(s) and waiting for the data to synchronize from the server to your Mac. It's better to grab all this data directly from the server rather than restoring it from backups, because the server almost certainly has fresher and more current versions of all the data, and restoring from backups may result in irritating collisions with live server syncing.