Markov property
A process has the Markov property when, conditional on its present state, its future evolution does not depend on the earlier history.
\[P(X_{n+1}=j\mid X_n=i,X_{n-1},\ldots)=P(X_{n+1}=j\mid X_n=i).\]Meaning
The present state must contain all information required by the model to determine probabilities of future transitions.
Biological modelling
A biological process is not automatically Markovian. If future behaviour depends on infection age, organism age or another unrecorded history variable, the state may need to be expanded to recover the Markov property.
Key idea. Memorylessness is a modelling property of the chosen state description, not a claim that biological organisms literally have no memory.