"Time Machine could not create a local snapshot to back up from." What do I do to fix this?

How much Free space does Disk Utility report for the "Data" (or "Macintosh HD - Data") volume?

Maybe try rebooting the computer to see if that helps.

Try running Disk Utility First Aid on the hidden Container. Within Disk Utility you may need to click "View" and select "Show All Devices" before the hidden Container appears on the left pane of Disk Utility. Even if First Aid says everything is "Ok", click "Show Details" to see if any unfixed errors are listed. If there are any unfixed errors listed, then try running First Aid while booted into Recovery Mode, otherwise you will need to erase the drive to perform a clean install followed by restoring from a backup.

To add a bit to @HWTech's excellent advice:

1. Verify Time Machine settings: Open "System Preferences" on your Mac and click on "Time Machine." Make sure the disk you are using for Time Machine backups is selected and that Time Machine is turned on. If the selected disk is not available or experiencing issues, you may need to select a different disk or troubleshoot the connection to the backup disk.
2. Restart your Mac: Sometimes, a simple restart can resolve temporary software glitches. Restart your Mac and try initiating a backup again with Time Machine.
3. Reset Time Machine preferences: If the issue persists, you can try resetting the Time Machine preferences. To do this, open "Terminal" (found in the Utilities folder within the Applications folder), and enter the following command: sudo tmutil associatedisk -a "/" You will be prompted to enter your administrator password. This command will disassociate the current Time Machine backup drive from your Mac. Afterward, go to "System Preferences" > "Time Machine" and select your backup disk again.
4. Check for file system errors: If there are file system errors on your startup disk, it can interfere with Time Machine's ability to create snapshots. Use Disk Utility to check and repair the disk. Open "Disk Utility" (found in the Utilities folder within the Applications folder), select your startup disk, and click on "First Aid" to perform a disk verification and repair if needed.
5. If you have an Intel-based iMac, reset NVRAM or PRAM: Resetting the NVRAM (non-volatile random-access memory) or PRAM (parameter random-access memory) can sometimes resolve issues with Time Machine. Restart your Mac and immediately press and hold the Option + Command + P + R keys until you hear the startup chime for the second time. Then release the keys and let your Mac restart normally.

DVHold wrote:

The Mac ((21.5-inch, Late 2015) Hard Drive shows 611 GB available on the 1 TB drive (11 GB used and 277.5 other volumes).

Sorry to nit-pick here, but "available" is either an unfortunate choice of words on your part or you are providing us with the wrong value since the "Available" value is meaningless to actually telling us how much actual storage space is available at this exact moment. The "Available" value is the amount of storage space that will at some unknown time in the future will be made available for use again....Apple previously used the more accurate & descriptive term of "Purgeable".

The "Free" space value shown in Disk Utility for the "Data" volume is the value you need to worry about since the "Free" space tells you how much storage space is available for use at this exact moment.

I did reboot the computer and also deleted a number of large video files from the hard drive, and the Time Machine back up seemed to work.

FYI, deleting files may not immediately free up space because the data may still reside in a hidden APFS snapshot which typically should be automatically deleted within a week. Again it is hard to say. Glad it worked this time.

But, I have so far been unable to verify that back up. The instructions for bringing up the "Verify Backups" command does not seem to work. (I'm using a PC keyboard and pressing the Alt button).

I won't be able to help here since I don't use Time Machine. Hopefully another more knowledgeable contributor will chime in here to assist with feature.

DVHold wrote:

The Free space and the available space, as displayed on my Mac HD in Disk utility, is the same: 610.33 GB out of the 1 TB drive.

Sometimes they may be the same value, but that is rare. The "Available" value is made up of the "Free" space + "Purgeable" items where "Purgeable" means the space will be released.

@Tesserax, thanks for chiming in here and expanding my understanding of TM.