Well, that's quite a bit of information but I'll give you a quick rundown.
In 1987 NEC (with a great deal of assistance from Hudson soft) released the PC Engine in the Japanese market. At the time the Famicom (Japanese NES) was far and away the dominant platform, while it took a bit of time to build up momentum the superior power of the PC-Engine started attracting developers. To cash in on the expanding success of the PC Engine, NEC made several variants of the system called CoreGrafx were developed with different colors and different configurations to attract more buyers as time went on. The PC Engine had fantastic potential and near arcade perfect versions of games like R-Type were being released, however there was one small problem, the PC Engine used HuCards (short for HUdson Cards since Hudson Soft was so heavily involved in the PC Engines development) which had limited space capability, to solve this problem and to allow this platform to grow NEC launched the PC Engine CD-rom, an add-on system that would connect to the PC Engine. This allowed the amount of space available for game design to jump from at the time an average of 2 megabits (there is 8 megabits in a megabyte) to 650 (and eventualy 700 and CD technology improved) Megabytes. Literally increasing the space capacity several thousand times. This allowed for developers to add CD quality sound, moving animation and other features that were simply impossible with the limited Space of the HuCard. The PC Engine CD-ROM used a system card that slid into the HuCard slot for memory.
In addition to CD-ROM technology NEC also designed another system called SuperGrafx. The Supergrafx was a more powerful platform that was backwards compatible with the PC Engine, unfortunately people weren't interested in buying a new system so shortly after the original PC Engine was released and the platform died quickly leaving only six released tiitles to its name.
NEC then released several upgrade cards for the PC Engine CD that upgraded its RAM (system Cards 2.0 and 3.0 which is known as Super CD).
In the meantime by 1989 NEC launced the TurboGrafx 16 (an aesthetically redesigned PC Engine with a different pin layout) in the United States. The system caught on fairly well and by its second year was quite popular fueled largely on the interest in the platforms mascot (Bonk), NEC then released a portable version of the PC Engine in both the US and Japan, known as the PC Engine GT in Japan and the Turbo Express in the US the expensive but fantastic quality handheld floundered due to its high cost.
In 1990 NEC test marketed the PC Engine in the UK as just "Turbo Grafx" sans the 16 that was part of the title of the system in North America. Apparently VERY VERY VERY few systems and games were produced. The European TurboGrafx was grey and opposed to black like the North American one, and the color of the sticker and logo was different.
That year NEC also released the Turbo Grafx 16 CD in North America with limited success due to its price and lack of software.
By 1992 NEC released a combination PC engine/ CD ROM 3.0 system called PC Engine Duo. This system saw a release in North Amerca as well as the Turbo Duo. In 1993 NEC released the Arcade card which added extra capability to the PC Engine, the Arcade card was planned for but was not released in North America. Another card called Game Express was designed and could only be played using a Duo since the card needed to be in the HuCard slot when playing CD games.
In 1994 NEC released the PC-FX which compared to its competitors the Playstation by Sony and Saturn by Sega was underpowered and lacking 3D capabilities. NEC was quickly relegated to a niche portion of the market limited almost exclusively to adult oriented titles and dating sims. The PC-FX died within 3 years of its release.
Does anyone else have any additions or corrections?