As their name already implies, their basis of operation is an internal gyro, spinning at very high speed (between ~ 10.000 - 18.000 RPM). The spinning gyro helps to errect the entire frame and gimbals* that surround the gyro and keep everything, including the instrument face (what you see) in place during flight. 

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What is a Gyro?

A Gyro is either a 'disc' or a 'wheel' spinning on a vertical or horizontal Gimbal*. 
Attitude Indicator = Gyro spins on a vertical Gimbal
Heading Indicator = Gyro spins on a horizontal Gimbal 
Turn Coordinator = Gyro (canted/angles at 30°) spins on a horizontal Gimbal  

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Due to the high rotational speed it is able to remain 'Rigid in Space', or in other words, stable in its plane of rotation. So, as we do normal Flight Maneuvers with our plane, the plane basically turns around the gyro. I know that this topic carries a lot of confusing words and terms. That why I find it much easier to understand by just seeing or watching an actual Gyro-Instrument, if that's possible whereever you do your Flight Training. Helps a lot, trust me.  However, to improve your imagination a bit at this point ...

There's a gyro for physical (arm) exercise, named PowerBall. Using it correctly, will make you aware of the inertia it can
have while spinning. The idea is to start its spin and then do and continue a certain motion with your hand to further
accelerate it and then keep its speed. So, the goal and strength (arm) training is to resist the force it creates.

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What does Rigid-In-Space mean?

It means that a gyroscope remains in a fixed position within the frame (gimbals) in which it is
spinning. The faster it is spinning the more stable it becomes. Again, as previsouly mentioned,
the gyro spins at around 10.000 - 18.000 RPM.
Newton's 1st Law (of Motion) says, 'a body in motion will maintain a constant speed (produced by the airflow of the vacuum pump) as well as its direction of motion unless affected by another force acting upon it'.

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Compare it to a top/spintop for kids. It needs acceleration (exerted by you) in order for it to spin. Depending on how strong you are able to make it spin, it will remain its (upright) position only for a given time. As rotation decreases, the spintop will slowly begin to stagger. After some point it lost all its spinning motion and literally just falls. <> The gyro in our instuments works in a similar fashion.

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What makes the gyros turn?

Two out of three instruments mentioned above (AI & HI) are driven by a Vacuum Pump.

This pump is again driven by the aircraft's engine. So, it draws air in all the way through tiny holes within the instrument's casing into ducts, which are engraved into the gyro wheel. Due to the strong suction of air produced by the pump, it is able to spin those gyros at the aforementioned speed range. This is also the reason, why we want to check for our 'vacuum readings' during our 'Before Take-Off' check. Usually a Checklist item. For the Cessna (172/152) this reading should be between 4.5 - 5.5 Now, the third gyro-instrument is the TC. This one however is electrically driven.

The reason being, in case of a vacuum pump failure, we have a back-up instrument for turning/banking indications. 

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Precession - What does it mean?

A force applied at a certain point is felt 90° (degrees) away from that point in the direction of the turn or turning motion. This is also the reason why the Attitude Indicator shows a slight bank/roll tendency when we pitch up and pitch up/down tendency as we increase/deacrease power. However, this effect is equalized again through the 'pendulous vanes' quickly. This is all very theoretical and not easy to understand. So, besides the picture you see below, my best suggestion is to just watch the related video from the link below.