An urn contains two red balls and one green ball.
One ball was drawn yesterday, one ball was drawn today, and the final ball
will be drawn tomorrow. All of the draws are "without replacement".
Suppose you know that today's ball was red, but you have no information
about yesterday's ball. The chance that tomorrow's ball will be red
is 1/2. That's because the only two remaining outcomes for this random experiment are "r,r,g" and "g,r,r".
On the other hand, if you know that both today and yesterday's balls were red, then you are guaranteed to get a green ball tomorrow.
This discrepancy shows that the probability distribution for tomorrow's color depends not only on the present value, but is also affected by information about the past. This stochastic process of observed colors doesn't have the Markov property.
Update: For any random experiment, there can be several related processes some of which
have the Markov property and others that don't.
For instance, if you change sampling "without replacement" to sampling "with replacement" in the urn experiment above, the process of observed colors will have the Markov property.
Another example: if $(X_n)$ is any stochastic process you get a related Markov
process by considering the historical process defined by
$$H_n=(X_0,X_1,\dots ,X_n).$$ In this setup, the Markov property is trivially fulfilled
since the current state includes all the past history.
In the other direction, you can lose the Markov property by combining states, or
"lumping". An example that I used in this MO answer, is to take a random walk $(S_n)$ on
the integers, and define $Y_n=1[S_n>0]$. If there is a long string of time points with $Y_n=1$, then it is quite likely that the random walk is nowhere near zero and that the
next value will also be 1. If you only know that the current value is 1, you are not
as confident that the next value will be 1. Intuitively, this is why $Y_n$ doesn't have
the Markov property.
For cases of lumping that preserve the Markov property, see this MSE answer.
Best Answer
Of course you have many application in "real life" problems, since all the motivation about the probability theory is to modeling this kind of problems.
You have stock markets applications how Brady Trainor said but not only this.
Stochastic process can be used to model the number of people or information data (computational network, p2p etc) in a queue over time where you suppose for example that the number of persons or information arrives is a poisson process.
Also in biology you have applications in evolutive ecology theory with birth-death process. In neuroscience, considering noise perturbations of ionic and chemical potential in neurons membrane.
In game theory, when you work with differential games for instance, which are a general framework for modeling many different "real word" problems in economy, computer science and others.
In optimisation and control of systems (stochastic control theory), were you typically model your uncertainty about the system interaction with the environment by stochastic process. Just to try to make it more concrete, automatic systems in general like for example the autopilot for cars ( you can obviously extend it for any other vehicle from mini robots to space shuttles)
In physics, more precisely in statistical physics formalisme and in complex systems.
It's just to try to give you a better idea about some possible applications and of course it's not an exhaustive list. I expect it can guide you for your research on the web.