The virial theorem is a theorem relating the kinetic and potential energies of a stable system as averaged over time.
2<T> = - Sum(over k=1..N) < F(k) * r(k) >
In a gravity-bound system, this amounts to the time-averaged kinetic energy equaling the negative of the time-averaged potential energy of the objects.
The theorem can be used to calculate of the average total kinetic or potential energy in complex systems: from one, you can get the other. In astrophysics, it is used to derive physical parameters (size, mass) of a system (stellar cluster, galaxy, galaxy cluster) from the velocities of the component bodies, which can be derived from spectral-line radial velocity measurements of the component bodies or from line broadening, if individual objects are not resolved.
The terms virial radius (sometimes virial length or virial scale) and virial mass are used in astrophysics in a couple of ways in regards to a system such as a galaxy or galaxy cluster:
Such values are often defined including everything up to the radius within which the average density (of the whole volume) is at some chosen multiple of the universe's average/critical density. Multiples of 100 and 200 are commonly used.
A virial temperature (TVir) is the temperature, e.g., of a gas, based on the kinetic energy of the gas particles made necessary by the virial theorem and other determined values related to the body of gas.