Very roughly speaking, you can think of a stochastic process as a process that evolves in a random way. The randomness can be involved in when the process evolves, and also how it evolves.
A very simple example of a stochastic process is the decay of a radioactive sample (with only one parent and one daughter product). Initially, it has some large number $N$ of atoms of the parent element. Over time, the number of such atoms decreases, always by $1$, but at random moments in time. The state of the system can be represented by $k$, the number of atoms of the parent element present at a given moment in time. Initially, $k = N$, but eventually, it will fall to zero.
In this process, when the state changes is random, but not how it changes. In other processes, such as a discrete-time random walk, when the state changes is deterministic, but how it changes is random. And there are other processes in which both when the state changes and how it changes are random.
Interestingly, in many cases, stochastic processes are used to model situations that may not have inherent randomness. For instance, Brownian motion is the result of forces that could, in principle, be determined precisely (if we ignored quantum mechanics). However, the number of objects in a normal system is so large that such an analysis would be intractable. Instead, we model the motion of objects using a stochastic process, and thereby obtain some insight into the behavior of such systems (for instance, the statistical behavior of a given particle over time) that we could not begin to with a deterministic approach.
The difference is really subtle. Citing Jeanblanc,Yor,Chesney (2009), they give the two following definitions:
The process X is a modification of Y if $\forall t$ $\mathbb{P}(X_t=Y_t)=1$.
The process X is indistinguishable from (or a version) of Y if {$\omega: X_t(\omega)=Y_t(\omega),\forall t$} is a measurable set and $\mathbb{P}(X_t=Y_t,\forall t)=1$
They moreover add the following relation: if $X$ and $Y$ are modifications of each other and are a.s. continuous, they are indistinguishable.
EDIT: There is an even more clear definition and explanation of the relation between the different definitions looking in Karatzas&Shreve (1998), p.2. Check it at this link.
Best Answer
A subordinator is a concept that applies to Lévy processes, where it is used as a stochastic time change which is itself an almost surely increasing Lévy process.
The thing is that the resulting process is still itself a Lévy process which is why it attracts interest. You can have a look at this paper. In general, every book on Lévy processes and stochastic calculus treats the subject, for example the book by Applebaum must have a section on this.