Sixteen-bit personal computer, based on an 8mhz MotorolaSixtyEightKay processor, developed by Atari in the mid 80s. Capable of displaying 320x200 with 16 colors, 640x200 with 4 colors, out of a total palette of 512 colors on a color monitor or TV. With a monochrome monitor, it could do 640x400 1bit. Used a Yamaha YM-2149 for sound. (A 3 voice square wave generator, a clone of the GI AY-3-8910, which was used in the ZX spectrum, the Amstrad CPC, and others, as well as a quite a few arcade games...) Atari licensed the GEM GUI from Digital Research. Was somewhat unique among the 16bit generation for having the operating system (Referred to as TOS...) entirely in ROM.

First released in 1985 as the 520ST, with 512k RAM. The 1040ST, with 1mb RAM was released in 1986, the first sub-$1000 computer with 1mb RAM.

Later milestone models include:

STe (1990) - Upped total color pallete to 4096, added 8bit stereo PCM sound output. Brought the ST somewhat more in line with the early Amigas. Used 30pin SIMMs for a maximum of 4mb RAM.

TT (1990) - Designed as a unix workstation. 32mhz 68030, with 68882 FPU. Could do 1280x960 on a special 19in monochrome monitor. With a color monitor, could do 640x480 16color, 320x480 256 color, and all ST/STe modes. STe sound chip. Internal SCSI hard drive. VME expansion slot, that can fit an ethernet interface. Four serial ports. A Unix SVr4 port by UniSoft was made available in the summer of 1992 and discontinued two months later. 

Falcon (1992) - Designed as a low cost "multimedia" machine. Its already slow 16mhz 68030 was crippled by a 16bit bus. Had a 32mhz 56001 DSP. Could do 640x480 8bit and 16bit color, plus all TT/STe/ST modes. (Well, except for the 1280x960 monochrome mode...) Internal laptop style IDE hard drive. Had the old single tasking TOS in ROM, but multitasking (and vaguely unix-like) MultiTOS was later available on disk. Thanks to the DSP and its 16bit stereo sound input and output, it kept the ST series popularity in the music field alive all the way into 1995, despite being dropped by Atari itself in 1993 so they could focus on the Jaguar game console. (Which bombed and pretty much killed Atari) However C-Labs licenced the falcon design and continued producing them until 1995.

The Atari ST didn't catch on much in the US in general, but it was pretty popular among musicians during the mid-late 80's, owing to its built in MIDI ports, (a ~$100 add-on for any other machine at the time) and overall low cost. Atari's slogan at the time was "Power Without the Price". It offered more RAM and better graphics than a Mac or PC of that era, for under $1000. But then Atari dragged its feet in releasing improved new models, and rapidly lost the market to the influx of cheap Windows 3.x/VGA based 386 clones in the early 90's.

The Atari ST did much better in Europe, especially in Germany for various reasons. But I'll let someone from Europe comment further on that.