Adding a second hard drive (/dev/sdb) to Debian Stable
I partitioned the new drive using parted.
I have the following diagnostics:
Rebooting, the system says that it cannot recognize the new hard drive.

??

You try to mount sdb1, but it has no filesystem.

You can either run or for more flexibility, create a logical volume from part of the space and build the filesystem there. That lets you expand or move filesystems later as needed.

When I typed
the system responded:
??

You have to create the filesystem on the partition -- /dev/sdbN (where N, presumably, == 1) -- and not on the entire drive.

HTH...

mkfs does not seem to work with the gpt file system.

I have tried
mkfs.ext4 /dev/sdb1
mkfs -t ext4 /dev/sdb

Both of those commands result in a corrupted file system.

Any chance you have gparted available? Could your version of parted be old?
Does the motherboard fully support the size of disk you installed and does it show in bios correctly?

GPT is a partitioning format, not a filesystem. And it's widely used. mkfs works very well with GPT disks.
How do you know it's corrupted?

The last one is wrong, you're formatting the whole disk again, not the partition, so overwriting all of the gpt info (which is in the beginning part of the disk, between block 1 and the start of the sdb1 partition itself).
Normally the first partition starts at about a MB (2048 blocks) from the beginning of the disk.

Problem solved:
I thank y'all for pointing me in the right direction.
That gave me a page of error messages, which threw me off completely. However, it was successful.

Then copy the "UUID=... " for /dev/sdb1 into fstab.
Success.

One minor problem: The directory (dolphin) lists the new drive as "primary", the alternative being "logical?". This could be confusing.

It's great that your problem is solved. I strongly doubt these were error messages, though. If they were, you should not ignore them.
I don't know if the concepts of primary and logical partitions make sense in the context of a GPT disk, but it is true that parted considers the partition primary, because you told it so:

I was working command line at the time, and the messages had something to do with buffer overflow. Not terribly surprising, since the disk is about eight years younger than the motherboard.

The gpt system apparently allows for being subdivided into a very large number of logical disks. Given that GPT can allow for something like 1000 petabytes, that is not too surprising.

One thing that I learned in passing is that portable storage devices should probably be formatted with ext3 instead of ext4 if you are going to be shuttling that drive from one machine to another. Ext4 wants to re-mount to the previous user.