I'm watching people go through a busy door. If I treat each coming and going as an event, I am left with a stochastic sequence. The sequence is not utterly unpredictable; I can analyze it and deduce trends (e.g. more people come in in the morning, and go out in the evening), but it is decidedly non-deterministic.
In this way, randomness acts as a sort of veil. The true shape of the sequence can be approached by statistical modeling, but it can never be fully grasped. There is something about it that remains surprising and unpredictable. This effect may be slight, but it can never be fully removed.
I have noticed that some people think of randomness as being some kind of synonym for "meaningless". For example, when a chain of causal reasoning reaches far enough back so that it gets to some random influence, the reasoning stops dead, unable to proceed. How it came to be becomes unanswered, and that lack of answer (or the seeming meaninglessness of the answer, such as "the coin flip came out heads") is extended back through the chain of reasoning to inflict the final result with a sort of senselessness. I'd like to dispell this perception.
Randomness does not represent a lack of meaning; it represents the addition of meaning.
Let's suppose we have a completely deterministic process, dependant on only a single parameter, some number that we provide. Depending on the process itself, it may seem incredibly boring [1], or very interesting [2]. From a computational perspective, however, if both the process and its parameter are known, the process becomes perfectly predictable. As stated at the outset, it's deterministic, so we can always compute the results to our satisfaction without ever being surprised by any of it.
[1]Like in the case where the process merely increments its parameter.
[2]Like in the case where the process produces pseudo-random numbers using the parameter as a seed.
When randomness is inserted into our process, it becomes non-deterministic. Each run of the same process with the same parameter may result in different outcomes. Their differences may be small or stay within some small deviation, or they may be large and grow as the processes are allowed to unfold. The degree to which the process tends towards the latter is the degree to which the process demonstrates chaos.
The point here is that randomness represents the intrusion of the unknown into a process. It adds uncertainty; it frees the systems it influences from the shackles of perfect predictability.
Uncertainty is relative to the observer, and as such ignorance is one source of randomness in a process. Consider our previous example of a deterministic process with a free parameter. If the process cannot be executed by the observer, then its results may appear to be random despite its known determinism. For example, pseudo-random number generators depend on this to produce their appearance of randomness (the lack of it from an objective perspective being captured in the "pseudo-" prefix).
Moreover, just because a process appears random doesn't mean that there is no method to its madness [3]. When I sat and observed people coming through a busy door, I understood that each of these people has their own life, their own routines, intentions, and plans. The order and spacing of events was a combination of these meaningful, yet unknown influences, and the multitude of ways in which they interacted with each other, producing the outcome that I witnessed.
[3]We are, each of us, deeply ignorant of the surrounding world. The vast majority of everything goes on beyond our understanding and awareness. In light of this, it may make more sense to think of ourselves as being cloaked in the veil of randomness, rather than thinking of bits and pieces of the outside world as being covered in that same veil.