Floppies disks were a way of storing data. They were square in shape, either 5" or 3.5" in size. The actual disc was inside, and was made from the same material you'd see in an audio or video cassette, only it in a disc shape.
Floppies didn't hold very much. The last popular format held 1.44MB (that's megabytes) For awhile another format, the Zip Disk, was popular. These looked like floppies, but were about twice as thick. They could hold 100MB, making them popular for offices since you could rewrite them, unlike CD.
Once CD burners became really cheap, the floppy basically started dying out. Every computer by that time at least had a CD drive, and software companies had long since stopped distributing software on floppies.
Then USB thumbdrives started to appear. These have quickly replaced the function that Floppies had, in terms of being a portable, easy to use method of carrying data between computers.
Most computers today don't even have a floppy drive. Floppies are just too old, slow, and small. Heck, I saw a box of Sugar Pops that was giving away a 1GB thumb drive. That's almost 700(!) floppy disks!
While people once used floppies to make backups, as you describe, nowadays people use USB thumbdrives or a USB hard drive if they have a lot of data they want to backup.
The point of backup is to have a 2nd copy in case something happens to your computer's main hard drive. Most USB hard drives come with automated backup software, making it very easy to automate the process. Otherwise you can also use a thumbdrive. When you insert the thumbdrive, it will show up like another hard drive on your computer. You can drag&drop files onto it as normal. The thumbdrive is nice because it's small, so you can take the files with you more easily.