Cryptanalytic computers are computers designed to be used for cryptanalysis. Modern cryptanalysis involves massive statistical analysis and multiple trial decryptions. These are possible only with automated equipment. This has been true since before WWII; Polish cryptanalysts designed and built automated aids in their work on Enigma traffic. Arguably, the first modern computer (digital, electronic, programmable (sort of)) was built for cryptanalytic work at Bletchley Park (the Colossus) during the war. More modern computers were important after WWII, and some machines (eg, the Cray-1) are reported to have had machine instructions hardwired in at the request of NSA.
Computers continue to be important in cryptanalysis well into the 21st century. NSA, in fact, is said to have the largest number of installed computers on the planet. Whether or not this is true in an age of Google computer farms and such remains publicly unknown.