I see the term 'high-beta' a lot, but cannot find a definition. When I type it into Google, I mostly see results about a Lockheed Martin fusion reactor project.....
The beta is plasma-$\beta$, the ratio of hydrodynamic pressure to magnetic pressure. It is useful because in extreme cases you can approximate the very complicated equations of magnetohydrodynamics. The equations simplify in the limits $\beta\rightarrow 0$ or $\beta\rightarrow \infty$.
For high-$\beta$ plasmas, $\beta\gg 1$. In this case, the hydrodynamic pressure supplies the dominant force. Basically, you can treat the system as a regular fluid that drags around the magnetic fields as it moves.
In low-$\beta$ plasmas, $\beta\ll 1$. In this case the magnetic forces dominate. You can solve the equations for the evolution of the magnetic fields, and the fields drag the fluid around with them.
Real plasmas are not usually so extreme. In fusion applications $\beta$ tells you how hard it is to magnetically confine the plasma.