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.