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The Nyquist criterion states that the number of clockwise encirclements of the (-1, 0) point in the G(jω)H(jω) polar plot equals the number of unstable closed-loop poles. A stable open-loop system with M unstable poles requires M counterclockwise encirclements for closed-loop stability. This provides both a graphical and analytical stability test.
From your study of frequency response, you know how to compute and plot G(jω) as ω sweeps from 0 to ∞ — the magnitude and phase of the open-loop transfer function at each frequency. The Nyquist criterion asks you to extend this to a full polar plot and watch what happens around one special point: (−1, 0) in the complex plane. That point is special because it represents the exact gain-and-phase condition for marginal instability in the closed-loop system.
The deep reason is the argument principle from complex analysis. Consider the closed-loop characteristic equation 1 + G(s)H(s) = 0, whose roots are the closed-loop poles. If you map the Nyquist contour — a large clockwise D-shaped path encircling the entire right-half plane — through G(s)H(s), the number of clockwise encirclements of the origin of 1 + GH equals the number of zeros of 1 + GH inside the contour (unstable closed-loop poles, Z) minus the number of poles of GH inside the contour (unstable open-loop poles, P). The same encirclement count around the origin of 1 + GH equals the encirclement count around (−1, 0) of GH itself, because they differ by a shift of 1 on the real axis. So: N = Z − P, where N is clockwise encirclements of (−1, 0).
For closed-loop stability, you need Z = 0 (no unstable closed-loop poles). Therefore you need N = −P, meaning P counterclockwise encirclements. If the open-loop system is stable (P = 0), closed-loop stability requires N = 0: the Nyquist plot must not encircle (−1, 0) at all. If the open-loop system has P unstable poles — as happens with some integrating systems or marginally stable plants — you need exactly P counterclockwise encirclements to cancel them. Crucially, this analysis works even when the open-loop system is unstable, which is something root locus and Bode methods handle less cleanly.
The connection to gain and phase margin from your prerequisite is direct. On the Nyquist plot, the gain margin is how much you can scale G(jω) before it reaches (−1, 0) along the negative real axis; the phase margin is how far the plot is from (−1, 0) at the unit-gain circle crossing. Both are geometric distances from the critical point. Nyquist provides the rigorous foundation that Bode diagrams approximate: Bode works well for systems without right-half-plane poles or zeros, but Nyquist handles the general case and makes the stability mechanism — encirclement of the critical point — explicit and countable rather than approximate.