The degeneracy pressure is the pressure exerted by a fermion gas even at zero temperature. The existence of a (potentially very large) pressure even at $T=0$ is counterintuitive and it represents a fundamentally quantum-mechanical effect.
The Pauli Exclusion Principle means that no two identical electrons can be in the same quantum state. If you imagine adding electrons one at a time to a finite region, with each electron falling into the lowest available energy state, you will soon need to be putting electrons into a states with substantial amounts of momentum, because the the low-lying states are already filled. So, even at $T=0$, many of the electrons in a dense electron gas are whizzing around at fairly high speed. It is the kinetic action of these moving electrons that is responsible for the degeneracy pressure.