Restoring individual items from your backup volume can be done in the Finder via drag and drop – simply find that item on the backup disk, then drag it back to your startup disk. If you're restoring more than a handful of items to your startup disk, configure a folder-to-folder task to restore specific folders from your backup disk to a specific folder on your startup disk.
If you're trying to restore system files, applications, or perhaps everything from your backup, proceed with the steps below to boot your Mac from the backup disk and then restore your backup disk to your Mac's internal hard drive.
Attach the Backup Volume to Your Macintosh
Attach the backup disk to your Mac using a USB, Firewire, or Thunderbolt cable.
Open the Startup Disk Preference Pane
System Preferences > Startup Disk
Select the Backup Volume
Click the padlock icon at the bottom of the window to authenticate, then select the backup disk that you would like to use to restore. Click Restart. This will reboot the system from your backup and allow you to clone the backup onto your main hard drive.
If you are unable to boot from your original hard drive, hold down the Option key as you start up your Mac. Your backup disk should appear as a startup disk option in the startup disk selector screen.
Prepare the disk that you're restoring to
Unless you're restoring just a handful of individual files, we recommend that you restore your backup to a freshly-formatted disk. See Preparing your backup disk for a backup of OS X for complete instructions on how to format the destination.
Open Carbon Copy Cloner
When your Mac has finished restarting, open Carbon Copy Cloner. Applications > Carbon Copy Cloner
Note: When you open CCC on your backup volume, CCC will prompt to guide you in setting up a restore task, in which case the instructions here are redundant. If you decline this offer, CCC will indicate that your regularly-scheduled tasks are suspended. If prompted, choose the option to leave your tasks suspended. Likewise, choose "Revert changes" if prompted to save your tasks.
Create a New Task
Click + in the TASKS header. Click Show Sidebar if necessary.
Name the new task.
Select the Source
Click on the dotted box under the Source heading to view available sources. Click to select your backup volume as the Source.
Select the Destination
Click on the dotted box under the Destination heading to view available destinations. Click to select the volume that you want to restore to.
Click the Clone button in the lower-right corner to start the resore task.
Reset the Startup Disk
After the clone is finished, choose Startup Disk from CCC's Utilities menu, then reset the startup disk to your original startup disk and restart your computer.
Test the Restoration
Launch a few applications and verify that your data is intact.