TITLE: Markov process vs. markov chain vs. random process vs. stochastic process vs. collection of random variables
QUESTION [27 upvotes]: I'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them.
According to Wikipeda:  

A Markov chain is a memoryless, random process.
A Markov process is a stochastic process, which exhibits the Markov property.
The Markov property is the memorylessness of a stochastic property.  
A stochastic process is a random process, which is a collection of random variables.
And finally, random variables are those determined by chance instead of other variables, which seems to mean explicitly that they are memoryless.

Thus, it seems that stochastic process, random process, Markov chain, and Markov process are all the exact same thing... which is a collection of random variables, which are memory-less, which means they exhibit the Markov property.

REPLY [2 votes]: The Wikipedia article on stochastic processes has been recently re-written to clarify this issue, among others. (Unfortunately the articles on specific stochastic processes such as Markov chains still need a lot of work.)
https://en.wikipedia.org/wiki/Stochastic_process#Markov_processes_and_chains
Usually a Markov chain is a specific type of Markov process. Unfortunately, as some have already mentioned, there is no agreed upon definition for a Markov process, except that either the state space or index set has to be discrete. For example, when people talk about a continuous-time Markov chain, they are using the convention that Markov chain refers to a Markov process with discrete state space. I would guess this is the more popular convention, but many use Markov chain to refer to Markov process with discrete time. I suppose, logically, if you accept both conventions then the terms "Markov process" and "Markov chain" become completely interchangeable. 
The term Markov chain is older, being used (in French) at least in the 1920s. Andrey Kolmogovov did pioneering in the 1930s work on Markov processes with continuous state space, but he didn't use the term "Markov process". William Feller decided to use that term in the 1940s.  
Markov processes and, consequently, Markov chains are both examples of stochastic processes.
Random process and stochastic process are completely interchangeable (at least in many books on the subject).  See relevant citations:
https://en.wikipedia.org/wiki/Stochastic_process#Terminology
Stochastic processes are, in general, not memoryless, but the term "memoryless" needs a precise definition. One could have, for example, a discrete-time stochastic process that depends on its states for the 3 previous times, but not all times, so in the long run it forgets all. Other processes only depend on the current state.