What are gain margin and phase margin, and why are they important?

Gain margin (GM) is how much gain can increase before instability, measured at frequency where phase is -180 degrees. Phase margin (PM) is how much phase can decrease before instability, measured at frequency where gain is 0 dB (unity). Both indicate robustness - how much plant variation the system can tolerate. Typical requirements: GM >= 6 dB (factor of 2), PM >= 45 degrees. Low margins indicate a system close to instability, sensitive to parameter changes. Phase margin relates to damping ratio in second-order approximation.