What are the union and intersection of events? I know what the union and intersection of sets are, but I don't understand how that applies to probability and events.(As in, why is the probability of the intersection of 2 events the probability of both events happening, and what sets are intersecting) A good explanation of events would also be helpful. 
 A: In probability we restrict our attention to a special set called the sample space. This contains all possible outcomes of an experiment. Now events are just subsets of this sample space. Given two events $A$ and $B$, their intersection and union are also events. They are defined in the same way they are for sets; namely
$$
A\cap B := \{x\in S \mid x\in A\ \text{and}\ x\in B\}
$$
and
$$
A\cup B := \{x\in S \mid x\in A\ \text{or}\ x\in B\},
$$
where $S$ denotes the sample space.
Thus if we want to find the probability that both $A$ and $B$ happen, we compute the probability of the intersection. Likewise for union.
A: We model an event as a set of outcomes. 
For example, let $\Omega = \{$ Raining, Sunny, Windy $\}$
Then we might want to ask "what's the probability of tomorrow being either Raining or Windy"?
Here, "being raining or windy" is an event that we could measure with a probability. And actually it is easy to see that this event could be represented by the set $A = \{$ Raining, Windy $\}$, which is a subset of $\Omega$. And you could also express $A$ as intersection or union of other events, for example, $A = \{$ Raining $\} \cup\{$ Windy $\}$. 
A: A sample space is the set of all possible outcomes.
An event is a subset of this sample space.   That is, an event is a set of outcomes.
Which is why we talk about their intersections, unions, and complements; may describe them as disjoint; and such.
The probability measure for an event is the expectation that the realised outcome is(or will be) a member of the event.
(The realised outcome is the one outcome that happened, occured, resulted, or other synonym.)

(Later you will learn that the sample space is coupled with a sigma-algebra and a probability measure to form a probability space.   The sigma-algebra of that probability space is basically the set of all measurable events.)
