recurrence (positive vs null) and transience
Start the chain at a state and ask: will it definitely come back, or might it wander off and never return? This single question splits every state into two fundamental types. A recurrent state is one the chain is certain to revisit (return probability 1); a transient state is one it might leave forever (return probability less than 1). Recurrence is about whether 'home' is guaranteed to be seen again.
From a transient state, the number of future visits is finite — the chain visits it some random number of times and then abandons it for good. From a recurrent state, the chain returns again and again, infinitely often. Recurrence splits further by how LONG the return takes. Positive recurrent means the expected return time is finite (returns happen at a steady rate); null recurrent means the chain is certain to return but the expected return time is infinite (returns happen, but ever more sluggishly). Null recurrence is a subtle in-between that lives only on infinite state spaces.
This classification decides a chain's long-run fate and the existence of equilibrium. An irreducible chain is all-recurrent or all-transient (it is a class property). A stationary distribution exists exactly when the chain is positive recurrent, and then the stationary probability of a state is one over its expected return time. Transient and null-recurrent chains have no proper equilibrium — probability either escapes to infinity or spreads out thinner and thinner forever.
The simple symmetric random walk on the integers (step left or right with probability 1/2 each) is null recurrent: it returns to 0 with probability 1, but the expected time to return is infinite. The same walk in 3 dimensions is transient — Polya's theorem says a drunkard on a line or plane comes home, but a drunk bird in space may fly off forever.
Recurrent: certain to return. Positive recurrent adds a finite expected return time; transient may leave forever.
On a FINITE irreducible chain every state is positive recurrent — null recurrence and transience can only appear when there are infinitely many states for probability to escape into.