I still haven't solved this issue, but I will update my questions.
I was running a normal update in Ubuntu 16.04, and suddenly the GNU GRUB screen appeared. I was updating via Ubuntu Software, and not from the terminal. I run 64-bit AMD, and I only have Ubuntu on this computer (no dual booting). The computer, which I purchased new, is under one year old. It is a Lenovo Ideapad 510.
During the day, I have seen at least two other people that appear to have had the same problem today, also both on 16.04:
https://ubuntuforums.org/showthread.php?t=2384382
I think, but cannot be sure, that I am having the same problem as these two. The person from the second link appears to have found a solution that does boot the system but with the some error messages. However, GNU GRUB appears to be quite complex and it is hard to know where to start in trying to learn about it. In other words, I do not understand what she did.
I did find this:
https://www.linux.com/learn/how-rescue-non-booting-grub-2-Linux
Which provided some beginner instructions on GRUB. I did manage to find where my kernel is (hd0,2). However the instructions here for booting from GRUB ultimately did not help me. It says that the vmlinuz version and the initrd version should be the same. However, it turns out that there are many different vmlinuz versions on my computer, and I don't know which one to choose. So then the above document says that: "On some Linux systems the current kernels and initrds are symlinked into the top level of the root filesystem:" so it is possible to try this:
grub> set root=(hd0,2)
grub> linux /vmlinuz root=/dev/sda1
grub> initrd /initrd.img
grub> boot
Well, I tried this. The terminal filled with different scripts(?), so I was first quite hopeful, but that process has now ended with the following curious last several lines:
No init found. Try passing init=bootarg.
[ 4.104041] usbcore: registered new interface driver usbhid
[ 4.104831] usbhid: USB HID core driver
BusyBox v1.222.1 (Ubuntu 1:1.22.0-15ubuntu1) built-in shell (ash)
Enter 'help' for a list of built-in commands
(initramfs) [ 7.925590] random: nonblocking pool is initialized
So there is where I am. Does anyone have any wisdom for me? I suppose one cannot exclude a hardware failure, but my gut tells me that this problem arose from that fateful Ubuntu update from this morning.
I see that some text is bolded in my question. I don't know why that is, and I apologize if this make my question hard to read.