Most first-time CCC users are looking to back up the Macintosh HD "startup disk". We walk through the steps for setting up that first backup task here.
Attach the backup disk to your computer
See the Choosing a backup drive section for additional advice on this subject.
Format the disk
Before you can use a new disk for a backing up your Mac, you must first initialize it with the correct format using Disk Utility.
Applications > Carbon Copy Cloner
When you open CCC for the first time, you'll be guided through your first task setup. If you prefer to not be guided, click the Tips button in CCC's toolbar.
Select the Source
Click in the box under the SOURCE heading to view available sources.
Select your startup disk from the menu of available volumes for the source.
Select the Destination
Click in the box under the DESTINATION heading to view available destinations, then select your new backup drive from the menu of available volumes for the destination.
Begin the backup task
Click Start. The first time you run a backup task, CCC will prompt you to grant it Full Disk Access, and also to authenticate so it can install its privileged helper tool. This helper tool is required to perform privileged tasks, e.g. to copy system settings and applications.
Congratulations - your first backup is in progress!
As CCC copies your files, you'll see a live performance chart like the one shown in the screenshot above. The white line tracks the write rate and the green line tracks the number of files evaluated per second. Hover your mouse over the chart to see the numerical values.
If you run the same backup task again, CCC will copy only the items that have changed. There's no special setting to achieve this behavior, simply click the Start button again or configure your backup task to run automatically on a scheduled basis.