event
An event is a collection of outcomes — formally, a subset of the sample space — that you single out because you want to know how likely it is. Where an outcome is one specific result, an event is a yes-or-no question whose answer is decided by which outcome actually happened. 'The die shows an even number' is an event; after the roll, you check whether the result fell inside the set {2, 4, 6}. If it did, we say the event occurred.
Because an event is just a set of outcomes, two extremes matter. The whole sample space Omega is an event — the one that always occurs, since some outcome must happen. The empty set, written with the symbol for nothing, is the impossible event — it contains no outcomes and so never occurs. In between sit all the interesting events. A single-outcome set like {4} is a perfectly good event too, and saying 'the event {4} occurred' is the same as saying 'the outcome was 4'. Probability is a number we attach to events: P(A) measures how likely the event A is.
Thinking of events as sets is the quiet engine of the whole subject, because it lets us combine questions with the language of sets. 'A and B both happen' becomes the intersection of the two sets; 'A or B happens' becomes their union; 'A does not happen' becomes the complement. So a sentence in plain English about chances turns into set arithmetic we can compute with, which is exactly why the algebra of events comes next.
Roll one die. Let A be the event 'an even number shows', so A = {2, 4, 6}. If the roll lands on 4, the outcome 4 is inside A, so event A occurred. If it lands on 3, A did not occur.
An event is a subset; it 'occurs' exactly when the actual outcome is one of its members.
Every event is a set of outcomes, but in infinite sample spaces not every subset can be assigned a probability — that subtlety belongs to the measure-theory field, not here.