As soon as the Atari 2600 was released the engineering team went back to design its eventual replacement. The newer design would be faster, have better graphics, and include much better sound hardware. Work continued throughout 1978, primarily working on the much-improved video hardware known as GTIA (the 2600 used a chip known as TIA).
However it was at this point in time that the home computer revolution took off in the form of the Apple II and TRS-80. Atari management saw this as a golden opportunity to re-purpose the machines, and started research on what would be needed to produce a workable home computer of their own. This included support for character graphics (something the 2600 didn't support), some form of expansion for peripherals, the BASIC programming language, and a keyboard.
All aspects of the machine were considered open to new solutions, and the machines ended up with its own BASIC, a wonderful peripheral system known as SIO, and a very powerful character/display driver known as ANTIC. Unlike the 2600 were the video was controlled solely by the TIA moving sprites (known as player/missile graphics in Atari lingo) around a colored background, in the new machines the ANTIC did most of the work drawing the screen (including characters), which the GTIA then colored and added the sprites. This separation of duties allowed both chips to be as powerful as possible, and the machine's graphics were the best on the market until the release of the Commodore 64 several years later.
Eventually they identified two "sweet spots" for such machines, the low-end Candy and high-end Colleen. Both ran at 2MHz, which made them twice as fast as most machines of the era; the Apple II and PET ran at 1MHz, the TRS-80 was at 2MHz but was actually about 1/2 of the speed due to its processor, the Zilog Z80's, design. The primary difference between the two was expandability, Colleen would include a number of memory slots, monitor output and a full keyboard, while Candy used a plastic "membrane" keyboard and didn't include any memory slots. Both machines were built like tanks, a side effect of meeting a FCC specification that was soon to be removed anyway.
The machines were brought to market in February 1979 as the 400 and 800, although they weren't widely available until late 1979. The names originally referred to the amount of memory, 4kB RAM in the 400 and 8kB in the 800. However by the time they were released the prices on RAM had started to fall, so the machines were instead released with 8kB and 16kB respectively, making the naming somewhat superfluous. In addition the 800's expansion system was not very usable, and the market seemed skeptical of the machines in general.
The 800 was rather complex and expensive to build, while the 400 didn't compete technically with some of the newer machines appearing in the early 1980s, so in 1982 Atari started the "Sweet 16" project to address these issues. The result was an upgraded set of machines otherwise similar to the 400 and 800, but much cheaper to produce due to the use of custom chips that replaced a number of chips from the earlier designs. Sweet 16 also looked to address problems with the 800 by adding a new expansion chassis as well. Like the earlier machines, the Sweet 16 would be released as the 1000 with 16kB and the 1000XL with 64kB.
But when the machines were actually released they came in only one version, the 1200XL. A number of problems in this machine, including a nasty bug in Atari BASIC made the machine a flop. This was quickly addressed in the hastily-completed 600XL and 800XL, which were largely identical to the original Sweet 16 specifications. However the 1200 was released at potentially the exact wrong time. By the time the new machines replaced it the Commodore 64 had already become the market leader, and Atari was unable to address this.
The final machines in the series were there 130XE and 65XE. These were really cut-down versions of the 600XL and 800XL in much cheaper cases, a result of Jack Tramiel's efforts to wring every dollar out of the platform before finally killing it. An additional 800XE was available in Europe, it was basically a 130XE with half the memory.
Atari's peripherals were named after the machines they were intended to be used with, so in general they have names like "410" and "1050". All of them used the proprietary SIO port, which allowed them to be daisy chained together into a single string. This resulted in far less "cable spaghetti" on the desk, but it also meant plugging in "standard" components like printers and modems could be difficult.
Atari also produced a number of other tape drives for use in eastern Europe where they continued to sell into the late 1980s due to their low cost. Some of these included various "high-speed" modes which made them almost as fast as early disk drives.
In addition to the list above Atari failed to release a huge selection of machines and peripherals that were otherwise completed. See the FAQ for details.