In aerodynamics, hypersonic speeds are speeds that are highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 (5 times the speed of sound) and above. The hypersonic regime is a subset of the supersonic regime.
Supersonic airflow is very different from subsonic flow. Nearly everything about the way an aircraft flies changes dramatically as it accelerates to supersonic speeds. Even with this strong demarcation, there is still some debate as to the definition of "supersonic". One definition is that the aircraft, as a whole, is traveling at Mach 1 or greater. More technical definitions state that it is only supersonic if the airflow over the entire aircraft is supersonic, which occurs around Mach 1.2 on typical designs. The range Mach 0.75 to 1.2 is therefore considered transonic.
Considering the problems with this simple definition, the precise Mach number at which a craft can be said to be fully hypersonic is even more elusive, especially since physical changes in the airflow (molecular dissociation, ionization) occur at quite different speeds. Generally, a combination of effects become important "as a whole" around Mach 5. The hypersonic regime is often defined as speeds where ramjets do not produce net thrust. This is a nebulous definition in itself, as there exists a proposed change to allow them to operate in the hypersonic regime (the Scramjet).