Is it (the "disk manager" of Wednesday, 17september2014) something that is finally going to let me “manage” making back-up disks for Windows 7?
No. There are different ways that disk space can be organized, & this GOTD deals with that sort of thing.
"This doggone Gateway has been telling me forever ~~ each time it’s rebooted ~~ to insert blank disks so it can do its backup (somehow it knows it’s not done, yet doesn’t let me)."
That's likely IMHO something that was added to your PC or laptop that's broken. Unless it's Windows backup [I've never used it so know next to nothing about win7's built-in backup], to fix it you'll probably need to uninstall that software, or at least prevent it from starting. You can find all sorts of advice & suggestions & how-to stuff on-line along with software to let you control what starts with Windows, but the best route I think is to find out what the software is, along with how to get rid of it. I'd start by using Google [include the make/model of your hardware] -- if it's a problem for you, it's been a problem for others too. Note that many systems come with all sorts of bloatware & may have a hidden disk partition(s) for storing backups, installation files etc. -- this may get into that territory, & you might find yourself reading discussions on how to get rid of all or most of that junk.
Now, the basics...
There are 3 types of backups... 1 is syncing often changing data -- usually stuff like e-mails & docs or whatever you might be working on. The example you'll see most often is in adverts about the cloud from Google & Microsoft, where a doc for example might live on-line, to be worked on using whatever device wherever you are.
2 is file backups -- at it's simplest it's combining several files in a .zip file that you'll store somewhere else. When you zip a file it usually gets smaller, so if you use a backup app to store a copy of all or some of the files/folders on your hard drive partition it usually takes up less space than the originals. Compressing [zipping] & uncompressing files uses your CPU's horsepower -- it takes longer & speed varies according to how fast & powerful your CPU is.
3 is a disk or partition image backup -- this is a copy of the raw data on the hard drive that isn't divided up into files & folders. It's fastest & doesn't require a lot of CPU power, but it works best when you're dealing with everything on the partition, since forcing it to deal with files & folders [to for example have it only back up part of a partition] slows it down tremendously. With image backups it's faster easier to back up everything, even if you're only interested in a portion of that data. Of the 3 types, only the image backup can restore your system disk so Windows boots.
Backup software...
There are 2 main types, those that operate in the background, normally apps for syncing &/or file backup, and those that do their thing while you go have a cup of coffee or tea, the image backup apps.
The relative speed of image backup software software depends on how fast your system can read from the hard drive it's backing up, & how fast it can write to wherever you're storing it -- the reverse applies when you're restoring a backup. That's one of the first things, if not the very first thing to think about when creating your backup strategy. If you've got $50 or less to spend, spend it on a hard drive rather than buying software. If you're thinking about buying an external drive to store your backups, know that USB 2, & *maybe* USB 3.0 would be much less usable. If you're thinking about optical discs, realize that restoring a backup is going to take a lot longer, though BD discs are the cheapest route if you'll save old backups.
From a usability standpoint, it's best to have a 2nd hard drive as your target connected either via SATA internally or eSATA externally. USB 3.0 can work well, if & when your hardware can use USB 3.0 to restore that backup. The same for firewire. That can be a tricky issue, since restoring a backup means booting to something besides your Windows install -- USB 3.0 has to be working in whatever you boot to or the data transfer will take a very long time via USB 2.0. That in turn depends on the electronics your PC/laptop uses for USB 3.0.
FWIW the last system [Windows 7] partition image backup I did takes up just over 71GB & took a little over a 1/2 hour to complete, writing to a 2nd internal drive -- both drives are slow to average speed. Restoration takes roughly the same amount of time.
Windows 7 comes with a reasonably good image backup program included [at least from what I've read from respectable sources]. The major alternatives AFAIK are made by Paragon, Acronis, EaseUS, & Aomei -- you should be able to get a version of all but Acronis for free. There are loads of apps for burning discs, CDs & DVDs & more recently Blu-Ray [BD]. The final results, the disc you burned, is normally the same regardless which software you used. The same thing can be said of image backup software -- the results should be identical whether you use win7 or one of the 4 brands I mentioned, since you're just copying raw data from one place to another, same as when you burn a CD/DVD/BD. Some people like one more than the others, &/or find that they have fewer problems running whatever brand, but as long as the software works properly there's no difference in the end results.