A "flippy disk" is a double-sided 5 1/4" floppy disk, specially modified so that the two sides can be used independently (but not simultaneously) in single-sided drives.
For operating systems that do not use the index hole in the disk to mark the beginnings of tracks, the "flippy" modification requires only a new write-enable notch to be cut. For those that do use index sync, however, a second index hole window must be punched in both sides of the jacket, and for hard sectored formats, an additional window must be punched for the sector holes. While cutting a second notch was relatively safe, cutting additional windows into the jacket was at great peril to the disk within.
A number of floppy disk manufacturers produced ready-made "flippy" media. As the cost of media went down, and double-sided drives became the standard, "flippies" became obsolete.
Floppy disks are read and written by a floppy disk drive or FDD, the latter initialism not to be confused with "fixed disk drive", which is an old IBM term for a hard disk drive.
The 3½-inch disks had, by way of their rigid case's slide-in-place metal cover, the significant advantage of being much better protected against unintended physical contact with the disk surface when the disk was handled outside the disk drive.
The 5¼-inch disk had a large circular hole in the center for the spindle of the drive and a small oval aperture in both sides of the plastic to allow the heads of the drive to read and write the data.