A Markov model is a description of a system that follows a Markov process. In a Markov process the next state of the system is a function of its current state and does not depend on where from and how it started. For example Brownian motion can be called a Markov process. The transition from current state to next state is described by probabilities.
So to conclude, a Markov model is a probabilistic model of a system that is assumed to have no memory.
In a Hidden Markov Model (HMM) the state(s) of the system are not known (therefore hidden). However, there are some functions that depend on the state and their outputs can be used to approximate the state of the system. There goes the definition- it is not that easy to explain how it is applied to different concepts (not just bioinformatics).
Strictly speaking this is off-topic. HMM is not a biological concept. It is a statistical/computational concept (it is like asking what is Dynamic Programming- which is the underlying concept behind BLAST algorithm).