Amiga :-) -
1. A series of home computers manufactured between 1985 and 1994. Famous for its at the time innovative architecture, custom chipsets, impressive graphics capabilities.
2.
Amiga Incorporated. The latest and current owners, licensees (from Gateway) and licensors of the Amiga intellectual property including the "Amiga" trademark.
Amiga one -
1. A trademark owned by
Eyetech Ltd. used for reselling third party hardware to the AmigaOS market. Currently covered by this trademark is the CHRP/POP-based
Teron series motherboards.
2. (obsolete) A planned but never produced
custom motherboard that was to be made for new versions of AmigaOS. Was to be designed by Escena and sold by Eyetech. An A1200 or A4000 mainboard would have been possible to attach for Amiga compatibility. Development stopped in 2001.
3. (obsolete) Any hardware complying to Amiga, Inc's arguably meaningless and vague "Zico specification".
Classic Amiga-
Term commonly used for referring to Amigas, as well as AmigaOS up to v 3.9.
Pegasos -
A series of CHRP/POP-based PowerPC motherboards made by Genesi. The Pegasos was superceded by the Pegasos II in 2003.
Blizzard PPC -
A PowerPC + 68k accelerator for the Amiga 1200.
AAA [board ?] -
The Amiga chipset standard that was to supercede the AGA (or "AA") chipset. It didn't happen.
c= -
Commodore Business Machines, CBM. The "C=" is an imitation of Commodores "chickenlips" logotype. CBM was the company behind the Amiga until its bankrupcy in 199?
Current status of amiga inc -
Insolvent and working from the employees' homes (allegedly unpayed), but has not yet filed for bankrupcy. Hardware production is entirely independent from AInc, and what's made for AmigaOS 4 is owned by
Hyperion.
Morph OS-
An OS that's delievered with the Pegasos hardware. It's API compatible with "classic AmigaOS" and includes a JIT 68k emulator for backwards compatibility with "classic Amiga" applications. Older MorphOS versions (?) also run on PPC-equipped Amigas.
The blue butterfly thingy-
Morpho. This butterfly is the MorphOS logotype.
The ball which has red and white things-
Comes from an old Amiga graphics demonstration. Currently one of the Amiga Inc. (and/or AmigaOS?) logotypes.
Why everyone attacks the butterfly with the boing ball :-) -
Uuuh?

Over definitions welcome
Well, something seems to be missing...
AmigaOS 4:
A rewritten and updated version of AmigaOS for third party hardware, initially the Teron/"AmigaOne" motherboards. There will also be a version for Blizzard/CyberstormPPC equipped Amigas. It's backwards compatible with "classic AmigaOS" plus its systemfriendly applications and will also include a 68k JIT emulator. Developed by Hyperion. A beta version for Teron owners has been announced to be released "some time after New Year" (in 2004, that is).
AROS:
"Amiga Research OS", an open source "classic AmigaOS"-API compatible OS for e.g. common x86 PC hardware