Consider a family of random variables $X_t,X_{t+1},...$ on a finite countable space $I$. The transition probability of moving from state $i$ at time $t$ to state $j$ is given by $P_{ij}^{t,t+1}=P(X_{t+1}=j|X_t=i)$. Normally we say that the family of random variables is time-homogeneous, which means that the law of the evolution of the system is time-independent (we can drop the time indices on the LHS).
For the irreducible property one can define an equivalence relation on $I$, and partition $I$ into communicating classes. If $I$ is a single class, then $P$ is called irreducible. In other words, it just means, that from any state, the state variable must be able to eventually reach any other state.
I am a little bit unsure now, but does that require time homogeneity, or is it possible to drop that and still have irreducibility?
A Markov chain that is not time-homogeneous can have a time-varying partition in communicating classes, which can possibly destroy the irreducibility property of the state space. Think for instance of a state space where there are two big components linked only by a single state in the middle. If the transition probabilities towards (or away from) this bottleneck state become zero at some point in time, then the state space is not connected anymore.
On the other hand, it can very well be the case that a Markov chain is not time-homogeneous, but still irreducible at all times (e.g. think of a Markov chain that is irreducible at time $t=0$ and such that all the transition probabilities that are non-zero at $t=0$ remain bounded away from zero at any time $t>0$.