A turbopump is a pump that sucks fluid in along the axis and expels it at high pressure and/or flow rate.
They work by having a turbine of some kind on a shaft connected to another turbine through which hot gas or another energy-rich fluid is past.
They are commonly used in rocketry to pressurise fuel so that it can be injected into the combustion chamber; they give good performance and can pump enormous volumes per second but are complicated to design, and only work well over narrow ranges of flow rates.
There's basically two difference sorts of turbopumps, centrifugal and axial. In centrifugal the fluid enters along the rotation axis and then is flung out sideways by the rotation of the blades.
Axial turbopumps have little blades sticking out from the central hub and rotate around, and force the fluid axially.
Rockets predominately use centrifugal pumps as they can produce higher pressures. Jet engines predominately use axial pumping to compress the air before the combustion chamber as it is more efficient. Interestingly the very first jet engines used centrifugal because it was easier to design; a few types of jets still use that design to this day.