What software/hardware is required to mine with an FPGA? I have been looking for a device that would allow me to mine Bitcoins without too much noise and heat. Something relatively small that I could keep in a shelf without worrying too much about it.

A rig with GPUs is too noisy so I looked into FPGAs, but I have no idea what these devices are. From what I understand, an FPGA is just a card (like a GPU) so I would still need to keep a computer on 24/7, right?

So do I need special hardware/software to connect an FPGA to my computer or does it work like a GPU?
You would need a computer to program the FPGA, obtain the work units, distribute them to the FPGAs, collect the shares, and submit them back. If you're not using a mining pool, you'll also need the computer to generate the work unit, assemble solved blocks, and submit them to the Bitcoin network.

The computer won't be doing that much work. So you can use a very low-end machine and optimize it for low power consumption and low noise.

Cablesaurus built the first commercially available FPGA Bitcoin miner. There is a lot of innovation going on and what's available is changing rapidly, but most of the current designs are very similar to this design.