Do commercial airliners have back up hydraulic systems to power the flaps, fin, etc...?

-I know they have that back up power generator thingy that drops out and catches the wind, but what about emergency hydraulic pumps, in case the actual motors/pumps that power the hydraulics fail...?Most commercial aircraft have one hydraulic pump on each engine and one electric pump to back up the engine driven pump.

So you would have "A" system engine pump, and "A" system electric pump. Then there is the other engine, which has another complete system "B" engine and "B" electric. There may also be 2 more standby pumps.

If you have a airliner with 4 engines, you have 4 engine-driven pumps, and then the electric ones to back them up. But 4 engined airliners are increasingly rare.

A lot of aircraft now have a "Left" system, driven by that engine and an Electric pump, a "Right"

system, also driven by the engine on that side, along with an electric pump, and then a "Center" system that is run by electric pumps only.

There are also valves that can be opened so that one system can drive the other one for a particular function. But this is left to the crew's discretion, as you don't want to have one system fail, then cross-connect to the next system and wipe it out as well. Typically, if one system fails, they ignore that system and run with what they have left.

There is a lot of variations in design. For a specific aircraft, you would have to look up that aircraft's hydraulic system and read how it works.

In modern aircraft manuals, the first section of each chapter in the Maintenance Manual is "Description and Operation" which tells how the system is designed, what powers what devices, and how the redundancies are arranged. So just reading this part of an aircraft manual would give you a good idea of how they do it for that particular aircraft..
Typically, aircraft will have 3 different hydraulic systems, each complete with its own pumps, pipes, accumulators, etc.

Each control surface or system will be powered by 2 different hydraulic systems (for instance the left aileron will be powered by a different system than the right aileron) so that a leak or other type of failure in one hydraulic system would still leave enough control capability.
There are all kinds of backup systems. Early jet airliners (including all current versions of the 737) in addition to having two independent hydraulic systems for the flight controls also had manual-reversion (like how you can still steer your car without power steering, but it is more difficult). Most modern aircraft have three or four independent hydraulic systems.



Flaps often have an electric backup system, but every aircraft is capable of landing without flaps in an emergency.



Landing gear can be blown down with emergency bottles, free fall, or be cranked down manually depending on the aircraft.



Brakes can use residual hydraulic pressure in the accumulator (again, like a car's power brakes) or use an emergency bottle as well.
"Power Generator Thingy"? Now that's an eloquent way of putting it! (not). Another point showing your general ignorance - one does not need to provide backup power to the fin, which is a fixed structural component. Perhaps you meant rudder, or the rudder trim servo tab. Sorry to be so harsh, but I'm encouraging you to ask more informed questions because I do not wish you to be the subject of ridicule. Thanks for the 2 points and an opportunity to educate.
Yes, either standby hydraulics or a backup electric motor. Not all aircraft have a RAT (Ram Air Turbine, the "back up power generator thingy" you refer to), some use an alternate hydraulic pump off the other engine, some use electric motors, etc. For example on the DC-3 you can manually pump the gear and flaps down, on the DC-9 the was a transfer pump driven by pressure from the other hydraulic system.
Some planes have a manual reversion system that allows the pilot to control the control surface with tabs. Instead of using hydraulic force they will switch to the aerodynamic force to help the pilot move the big control surfaces.
As an example, in a 747, if you lost hydraulics, you extend gear and flaps with electric motors. The 747 have plenty of back-up systems - And that is an airplane that dates back to the late 1960s...!