That's the beauty (and sometimes the curse) of hardware design: it doesn't need a processor to run. You essentially describe an electronic circuit, and "program" (configure would be more appropriate) an FPGA to act like the circuit you described. A circuit can be as simple as you want (like a sequential counter) or very complex (people have implemented entire H.264 and HEVC decoders in hardware).

This is so powerful that in fact you can even describe your own processor, and prototype it on an FPGA for testing.

> I'm actually generating nice Verilog specifications. Can I apply it somehow if I'm not working in Intel or something? What can I do with it?

Throw it into an FPGA

And it would be significantly faster than if I would concentrate on optimizing the code for, say, executing it on GPU?

Perhaps. Perhaps not. If you have an algorithm that is better implemented in hardware, you might well see better performance executing it on a GPU (for example, if you want to mine bitcoins).

Really, FPGAs are meant for prototyping hardware; you synthesize your ASIC onto the FPGA and check it for functionality.

Of course, there's a reason you're making that ASIC.

Or you don't do an ASIC and ship the FPGA. That's common as well (for the big manufacturers it's cheaper to do an ASIC, for the not so big, it's more common to sell the FPGA)

Depends on what you want to do.

It's not about faster, it's about the application, you can't plug a PC with a GPU to solve all problems.

FPGAs weren't created for bitcoin mining.