irreducibility
Imagine a country where you can travel from any town to any other town, perhaps with several legs in between. Nothing is walled off; the whole land is one connected territory. A Markov chain with that property is called irreducible: from every state you can eventually reach every other state. It is the cleanest, friendliest kind of chain, and the condition under which the prettiest theorems hold.
Formally, a chain is irreducible if all its states communicate with one another — there is just a single communicating class containing everything. Equivalently, for every pair of states i and j there is some n with P^(n)(i, j) greater than 0: a positive-probability path connects any state to any other. There are no isolated pockets, no states you can get stuck unable to leave, no regions you can never reach.
Irreducibility matters because it removes the bookkeeping headaches of multiple classes and lets the chain behave as one coherent system. Combined with positive recurrence it guarantees a unique stationary distribution; add aperiodicity and the chain converges to that distribution from any start. Most of the clean convergence and equilibrium results you will meet — the fundamental limit theorem, the ergodic theorem — assume irreducibility as their starting hypothesis.
The weather chain with rows (0.8, 0.2) and (0.4, 0.6) is irreducible: sun reaches rain (prob 0.2) and rain reaches sun (prob 0.4), so the two states communicate. But add a state 'snow' that you can enter from rain yet never leave, and the chain stops being irreducible — snow is a trap, a separate closed class.
Irreducible means one big communicating class — every state reachable from every other.
Irreducibility says any state can reach any other EVENTUALLY (in some number of steps), not in a single step. It is about reachability over time, not about every transition having positive probability.