I am unable to understand these concepts clearly. I am getting lost while reading about these. I have googled about these concepts but still nothing helped me to understand these concepts.
According to my understanding so far, a Markov process is where in which the future states depends on the current state but not all previous state. And a Markov chain is where the process contains finite or countably infinite number of states.
Please explain me to understand the poisson process and How can we relate a Markov process to poisson process. Thank you very much in advance.
A Markov process is a stochastic process where the distribution of a state $X_t$ conditioned on $X_s$ at various times less than $t$ only depends on the state at the last of those times (and possibly $t$ itself). Thus the process forgets all past states that it was in as it goes.
Markov chain is not a standardized term. All usages of the term include discrete time finite state Markov processes. Whether continuous time or countable states can be included varies by author. Continuous state processes are usually not included.
A Poisson process is a continuous time Markov process on the nonnegative integers where all transitions are a jump of $+1$ and the times between jumps are independent exponential random variables with the same rate parameter $\lambda$.