Oof: I wrote all of
The standard big results here are: (1) there is a critical probability, usually denoted p_c, such that if you declare squares open with probability p < p_c, the probability of an infinite open component is zero, while with p > p_c it is 1. In dimension two, it's even known that p <= p_c is the probability zero condition; what happens at p_c exactly is known for some other lattices, but I don't know the full story.
... but I neglected to mention that Robert Ziff's 1992 paper "Spanning probability in 2D percolation" (Phys. Rev. Lett. 69, 2670–2673 (1992)) calculated that p_c for the 2d square lattice is 0.5927460 ± 0.0000005 --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.