Wet aging beef is simply the process of storing it in a vacuum sealed plastic bag at near freezing temperatures. The first use of this system for meat was the Cryovac sealing of turkeys in the fifties. With improved plastics and sealing technology in the seventies vacuum sealing began to take off and has become the dominant form of ageing as it far more economical.
Dry aging has been in use for at least hundreds of years and in its most basic form simply means hanging meat in air. With modern advanced techniques temperature, air flow, humidity and unwanted microorganisms are carefully controlled, but the basic idea is still just hanging the meat out to dry.