the maximum principle for harmonic functions
Stand inside a metal plate at steady temperature and ask: can the hottest spot be somewhere in the middle? The answer is no — the heat would flow away from any interior peak until it flattened. The warmest and coolest places must sit on the boundary. This is the maximum principle (and, by the same logic, the minimum principle) for harmonic functions: a nonconstant harmonic function attains neither its maximum nor its minimum at any interior point of its region.
More precisely, let u be harmonic on a bounded region D and continuous up to its boundary. Then the maximum and the minimum of u over the closure of D are both achieved on the boundary of D. If a maximum (or minimum) is reached at an interior point, then u must be constant on the whole connected region — that is the strong form. The engine behind it is the mean-value property: at any interior point z_0, the value u(z_0) equals the average of u over a small circle centered at z_0. If u(z_0) were the largest value around, the average of the neighbors could not exceed it, so all nearby values must equal u(z_0); repeating this spreads the maximum out and pins u as constant.
The payoff is uniqueness for boundary-value problems. If two harmonic functions agree on the boundary of D, their difference is harmonic with boundary values zero, so by the maximum and minimum principles the difference is squeezed between 0 and 0 throughout — it is identically zero. Hence a harmonic function is completely determined by its boundary values, which is exactly what makes the Dirichlet problem well posed. A caution: this needs the region to be bounded; on unbounded regions a harmonic function can grow without ever attaining an interior maximum, so the principle must be stated with care (one uses a behavior-at-infinity hypothesis).
On the unit disk, u(x, y) = x is harmonic; its largest value on the closed disk is 1 and its smallest is -1, both attained on the boundary circle (at the points 1 and -1), never inside.
The harmonic function x reaches its extremes only on the boundary, exactly as the principle demands.
The principle is about real harmonic functions and their values; the analogous statement for holomorphic functions is the maximum-modulus principle, which is about |f|, not f itself (an interior maximum of Re f is also forbidden, but Re f can be negative — there is no 'minimum-modulus' unless f is nonvanishing).