An MSX-ENGINE chip is a specially developed integrated circuit for home computers that are built according to the MSX specifications. Generally, such a chip combines the functions of many separate, older/simpler chips into one. This is done to reduce required circuit board space, power consumption, and (most importantly) production costs for complete systems.
The first MSX-Engine chip, the T7775 operated next to a standard Zilog Z80-clone chip, the main CPU of the system, but most later versions of the engine also included the Z80 (clone) CPU in a same single chip package. The S-1990, is a special case, as it's not really an MSX-Engine, but a chip that was used as "glue logic" between the MSX-engine and an external R800 CPU.
The T9769 is used in MSX 2 computers, while in MSX 1 computers mostly the T7775 and T7937 are used. You can also find the S-1985 and S-3527 in these systems. After the MSX 2 generation (from MSX2+ onwards) Toshiba took over the complete production of MSX engine chips. The last generation of MSX, the Turbo-R used the NEC S-1990 "TurboR bus controller" together with a R800 CPU. MSX engine chips from Yamaha were mostly used in MSX-computers from Sony and Philips, while the Toshiba chips were mostly used in computers from Sanyo and Matsushita (Panasonic/National).