ATA Packet Interface (ATAPI) is a protocol that has been applied to Parallel ATA and Serial ATA that allows a greater variety of devices to be connected to a computer than ATA would allow.
ATA was originally designed for, and worked only with hard disks and devices that could emulate them. The introduction of ATAPI (ATA Packet Interface) by a group called the Small Form Factor committee (SFF) allowed ATA to be used for a variety of other devices that require functions beyond those necessary for hard disks. For example, any removable media device needs a "media eject" command, and a way for the host to determine whether the media is present, and these were not provided in the ATA protocol.
The Small Form Factor committee approached this problem by defining ATAPI, the "ATA Packet Interface". ATAPI is actually a protocol allowing the ATA interface to carry SCSI commands and responses; therefore all ATAPI devices are actually "speaking SCSI" other than at the electrical interface. In fact, some early ATAPI devices were simply SCSI devices with an ATA/ATAPI to SCSI protocol converter added on. The SCSI commands and responses are embedded in "packets" (hence "ATA Packet Interface") for transmission on the ATA cable. This allows any device class for which a SCSI command set has been defined to be interfaced via ATA/ATAPI.
ATAPI devices are also "speaking ATA", as the ATA physical interface and protocol are still being used to send the packets. On the other hand, ATA hard drives and solid state drives do not use ATAPI.
ATAPI devices include CD-ROM and DVD-ROM drives, tape drives, and large-capacity floppy drives such as the Zip drive, Magneto-optical drive and SuperDisk drive.
The SCSI commands and responses used by each class of ATAPI device (CD-ROM, tape, etc.) are described in other documents or specifications specific to those device classes and are not within ATA/ATAPI or the T13 committee's purview. One commonly used set is defined in the MMC SCSI command set.