FPGAs (Field-Programmable Gate Arrays), are chips with programmable circuitry. Rather than spending millions of dollars to fabricate a custom chip (or ASIC--Application-Specific Integrated Circuit), you can program your custom logic onto an FPGA almost instantly. While ASICs can be more deeply optimized, FPGAs are far more accessible, and are even available in the cloud.
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Verilog is the most popular RTL (Register Transfer Level) language for modeling digital circuits (err, well, analog too) so they can be turned into physical hardware or programmed onto an FPGA by some really sophisticated software and tooling. Other Hardware Description Languages (HDLs) aim to address the limitations of Verilog and enable design at a higher level.
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