The first half of this phrase is the more wibbly-wobbly part of the name. Pilot wave is an interpretation of quantum mechanics in which a wave field guides a particle, and the particle generates the wave as it goes. Essentially, instead of things like electrons and protons being both particles and waves at the same time, they are made up of two parts, a particle and a wave, working together.
Hydrodynamics is, as the name implies, the study of fluids (hydro) and how it moves (dynamics). It also includes the study of other objects moving through and interacting with fluids.
Pilot Wave Hydrodynamics is a Hydrodynamic real world model of Pilot Wave theory, where a low-viscosity bath is shaken up and down at about 60Hz. Then a silicone droplet is placed on top of the bath, hitting the surface and bouncing off of it instead of coalescing. Each bounce creates a standing wave on the oil that changes how the droplet bounces off it next time around. An intresting effect that arises from this is path memory, where the former location of the droplet effects the wave field, and thus how it moves in the present. Another is that the wave field is non-local, so while the droplet only passes through, say, one slit, the waves pass through both and affect its trajectory.
Quite a few people have done PWH experiments before me, and they also had really nice cameras. Dan Harris has given lots of advice for building my setup, and he also took great videos with stripped lighting to show wave topography.
I don't have much more to say, except that I hope I can make something that awe-inspiring.