FPGA computing with Debian and derivatives
FPGA  (Field-programmable gate array) can be programmed to perform a particular computation in hardware. And when the array is large enough, it can perform many such computations in parallel. This resembles the execution of code on the GPU, just that the GPU can other than the FPGA not be changed in its functionality.
The flexibility of FPGAs makes them come in various flavours. They are usually coming on PCI or USB-pluggable boards, or they occupy a CPU slot. And when on a separate board, they may have some extra intelligence with them, which may be capable to run Linux. And there are yet again other setups, that let an FPGA emulate a CPU ... confusion.
The FPGA vendors coming to mind first are Altera, Lattice and Xilinx . When aiming at just getting some problem with FPGA solved with that machine attached to the net or to the USB port in a Debian compatible manner, there may be several options [3,4,5].
Debian had the ZTEX  in its distribution, removed since there was only limited gain over the use of upstream's pages, see ztex-ezusb:
- their boards are Open Source hardware, prepared for IO, application acceleration,
the company regularly contributes to http://OpenCores.org, and
- their API is Free.
While the ZTEX boards use Xilinx FPGAs themselves, that vendor supports Linux also directly, also for the boards to run the OS themselves. See FPGA/Xilinx for details.
FPGA and the DFSG
FPGA commonly take their "layout"/"program" from some memory local to the FPGA. The program executed on the FPGA then does not touch that configuration but moves data to it and reads the results. The code of the FPGA is prepared in some Harware Description Language (HDL), where VHDL and Verilog are the ones seen most often. All would be fine if the tools to prepare the initial bitstream for the FPGA were Freely available. And for the better FPGA they are not even free as in beer. There are HDL emulators, but that is not fun. So, as a consequence, we can have the bitstreams in Debian non-free, and all tools relying on FPGA technologies only in contrib.
Anybody knowing better please correct this paragraph.
FPGA and the community
To employ FPGA technologies has enormous ecological advantages - the computation is so much faster with so little energy used for the acceleration. And the bitstreams produced are produced for the FPGA, in complete ignorance of the desktop operating system communicating for it. Hence - we should find ways to collaborate all across the Open Source landscape. Most advanced are
http://www.ohwr.org/ - CERN's Open Hardware site
where there is yet nothing about application acceleration, though.
-  please add your Debian-compatible FPGA resource here