the z-transform
Digital signals come as sequences: a list of numbers x_0, x_1, x_2, ... sampled one tick at a time. The z-transform is to discrete sequences what the Laplace transform is to continuous functions — a way to package the whole sequence into a single function of a complex variable, so that the messy business of delays and feedback becomes ordinary algebra. It is the native language of digital filters, digital control, and discrete-time systems.
The definition is a power series in 1 / z: X(z) = sum over n of x_n z^(-n). For a one-sided (causal) signal the sum runs from n = 0 upward, and it converges in a region of the complex plane of the form |z| > R, the region of convergence, which is an annulus or the outside of a disk. The variable z carries the meaning 'advance by one sample,' so a one-step delay of the sequence simply multiplies the transform by z^(-1) — exactly the way differentiation multiplies a Laplace transform by s. This single fact turns a difference equation (the discrete analogue of a differential equation) into a polynomial-ratio equation X(z) = (something) / (a polynomial in z), which you then invert by partial fractions, by reading off a known series, or by a contour integral x_n = (1 / (2 pi i)) times (integral over a circle in the region of convergence of X(z) z^(n-1) dz) evaluated as a sum of residues.
The z-transform is the complex-analytic backbone of all discrete signal processing. Its poles and zeros sit in the complex plane and their positions relative to the unit circle |z| = 1 decide everything: poles inside the unit circle mean a stable, decaying response, poles on or outside it mean ringing or blow-up. (This is the discrete mirror of the Laplace left-half-plane stability rule.) A genuine subtlety to keep honest: the same algebraic formula X(z) can correspond to different sequences depending on which region of convergence you pick — a causal right-sided signal versus an anticausal left-sided one — so the region of convergence is part of the answer, not an afterthought.
The sequence x_n = a^n for n >= 0 has X(z) = sum over n >= 0 of a^n z^(-n) = 1 / (1 - a z^(-1)) = z / (z - a), valid for |z| > |a|. Its single pole sits at z = a; it is inside the unit circle exactly when |a| < 1, which is precisely the condition for the sequence a^n to decay rather than grow.
A pole inside the unit circle (|a| < 1) means a decaying sequence — discrete stability.
Do not confuse the unit circle with the imaginary axis: for Laplace the stability boundary is Re(s) = 0, but for the z-transform it is |z| = 1, because the sampling map z = e^(s T) wraps the left half s-plane onto the inside of the unit disk.