PC-DOS 1.44M format
The PC-DOS 1.44M format (3 1/2", double sided, high density) was the high-density counterpart of the PC-DOS 720K format for 3 1/2" floppy disks, storing twice as much data due to use of a different media surface capable of handling a higher density of data. It had 80 tracks per side, with 18 sectors per track, and 512 bytes per sector. Data was stored with MFM encoding. The disk turned at 300 RPM.
These disks were generally used with FAT12 file systems under the MS-DOS or PC-DOS operating system. High-density disk drives could handle both the new 1.44 MB format and the old 720K format, though there could be compatibility issues in reading 720K disks on low-density drives after they were written to with a high-density drive, even though the writing is done in an emulation of the old format, due to the different drive head on the newer drives. (This sort of incompatibility might, however, not have been as common as that between the high and low density 5 1/4" disks.) The drive could tell which type of disk was inserted because high-density disks had an additional punched hole on the upper-left corner (both high and low density disks used a write-protect hole in the upper-right corner of the casing (when seen from the label side) which could be opened or closed depending on whether you wanted to protect the disk from writing). Users occasionally punched holes in low-density disks to attempt to use them in high-density mode, but this did not work very reliably; the media was not designed for this.
This 1.44 MB format was the highest capacity floppy disk format in common use, but several higher-capacity formats also existed (but didn't catch on much) under the label "extended density". An IBM Extended Density format supported in DOS 7 and OS/2 Warp 3 could get 1.68 MB or 1.72 MB on a standard high density disk (by using 21 sectors in 80 or 82 tracks) and was used for system software distribution, on disks whose first cylinder uses standard formatting to allow room for a README file accessible without support of this format (see Wikipedia article). A different sort of "Extended Density" required a different type of disk media for the PC-DOS 2.88M format, used on high-end PS/2 models.
In the late '80s and early '90s, it was common for desktop PCs to have both 5 1/4" and 3 1/2" disk drives in order to be compatible with all software and data, which might be distributed on either format; by the 1990s these were usually high-density drives supporting the 1.2M and 1.44M formats. Often the 5 1/4" drive was drive A, and the 3 1/2" one was drive B. Later PCs, however, were more likely to have only a 3 1/2" drive, set up to respond to both drive letters. Eventually, PCs stopped having floppy disk drives altogether as other data storage and transfer media took over.
3 1/2" disks are actually 90 mm wide, but are almost universally referred to as "3 1/2 inch" disks even in countries that use the metric system.
It's easy to assume that doubling 720K gives you 1.44M, but unfortunately it's more complicated than that. The actual effective capacity of a "1.44M" floppy is 1440 KiB, where a KiB is 1024 bytes. This works out to 1.47456 MB (where a MB is 106 bytes), or 1.40625 MiB (where a MiB is 220 bytes). It's not 1.44 MB, for any sensible definition of "MB".
See also Floppy disk#Other links and references.