A distinction I don't see made often enough is between what I call randomness and ignorance. Roughly, every expression of uncertainty is either about "where in the universe am I?" or "what is the universe like?" (or both). The former is the domain of randomness, the latter of ignorance.
Suppose you roll a die. You know that you're in a situation where you've just rolled a die, and that, in roughly 1/6th of the situations where one has just rolled a die, the die will come up a three. Thus, your uncertainty about the die roll is random.
Suppose you're wondering whether or not an omnipotent and immortal being exists. Whatever the answer is, it is the same every time someone is asking this question. Thus there is no randomness involved, but you are ignorant of what the answer is (though you might have a hunch).
Often, your uncertainty will have components of both. Suppose someone hands you a die, you roll it five times, and every time you roll a three. You can probably guess that the die is biased. But why? One way to answer this question is that, in most of the situations where one is handed a die and rolls it five times and gets the same number five times, the die is biased. You are ignorant of exactly how often this is the case, though. It could be the case every 9 out of 10 times this happens, or perhaps every 8 out of 10 times. Now suppose you knew that it was 9 out of 10 times. Then it would still be random whether you are in one of the 9 cases, or in the tenth.
Next up: Reference Classes for Randomness