the critical probability
The critical probability p_c is the precise location of the percolation phase transition: the parameter value at which the system switches, abruptly, from having no infinite cluster to having one. It is the central number of the subject. Everything interesting — the scaling laws, the fractal critical clusters, the conformal invariance in two dimensions — lives in a vanishingly thin window around p_c. Identifying its value, and proving that the transition there is genuinely sharp, is the core program of percolation theory.
The clean definition is p_c = sup{ p : theta(p) = 0 } = inf{ p : theta(p) > 0 }, the threshold of the percolation probability. For p < p_c (the subcritical phase) the origin's cluster is almost surely finite; for p > p_c (the supercritical phase) it is infinite with positive probability, and an infinite cluster exists almost surely. A first nontrivial fact is that p_c is strictly interior: 0 < p_c < 1 in every dimension d >= 2. The upper bound p_c < 1 comes from a Peierls-type argument — count self-avoiding paths from the origin and show that at p close enough to 1 the expected number of open paths to distance n grows, forcing an infinite cluster. The lower bound p_c > 0 comes from a counting bound on contours or paths: the expected number of open self-avoiding paths of length n from the origin is at most (mu p)^n where mu is the connective constant, so for p < 1/mu this sum is finite and the cluster cannot be infinite. In one dimension the model is degenerate: any single closed edge cuts the line, so p_c = 1.
A more recent and powerful insight is that the transition is SHARP, not gradual. The Menshikov and Aizenman-Barsky theorems (and Duminil-Copin and Tassion's modern proof) show that the threshold defined via theta(p) coincides with the threshold for exponential decay of cluster sizes: there is a single point p_c, below which connection probabilities decay exponentially and above which there is an infinite cluster, with no intermediate 'critical-like' regime. So the many a priori different thresholds one might define all collapse to the same p_c. Honest caveat: except in special exactly-solvable cases (Z^2 bond, triangular site, trees), the numerical value of p_c is not known in closed form — it is a lattice-dependent constant we can only estimate.
Site percolation on the triangular lattice has p_c = 1/2, provable by a self-matching duality argument; this is the model for which Smirnov proved conformal invariance. By contrast, bond percolation on the simple cubic lattice Z^3 has p_c approximately 0.2488, a number obtained only by high-precision Monte Carlo and series expansions, with no known formula.
p_c is exact in a handful of self-dual lattices and merely numerical everywhere else.
Do not conflate p_c (where an infinite cluster first appears) with the value of theta(p_c) (the density at criticality). Whether theta(p_c) = 0 is a separate, harder question, open in dimension 3 as of the early 2020s.