How does conditional probability relate to mathematical operators? How to know when to define events as conditional? I noticed in my introductory statistics course that most operators like AND/OR relate to mathematics operators like multiplication and addition. Is there any relation between conditional probability to mathematical operators?
My follow up question for this is: I'm struggling to determine when to define an event as a conditional probability event or when to define it as A and B have occurred. I'm not sure why this is not making sense for me, but it'd help to know a better way to understand when and how to define conditional probability.
 A: Probability theory is a branch of mathematics, so it is a "mathematical operator". It is defined in terms of joint probability of two events $A$ and $B$ occurring together
$$
P(A\; \mathrm{and} \;B) = P(A, B)
$$
and the marginal probability $P(B)$,
$$
P(A | B) = \frac{P(A,B)}{P(B)}
$$
A: The simplest way to understand conditional probability at an introductory level is to undertake exercises where you relate probabilities to areas and relative-areas in a Venn diagram.  There are many resources available that teach conditional probability in this way, but I will give a simple outline here.  Consider a Venn diagram with two events $A$ and $B$ represented as interlocking circles (picture taken from this related question), and suppose we treat the areas as being equivalent to the probabilities.  The probability of $A$ is the area of the purple circle and the probability of $B$ is the area of the blue circle.  The conditional probability of $A$ given $B$ would be the relative size of the intersection where both these events occur (the dark-blue area in the middle) relative to the area of the blue circle.

If you do a quick search online you will find hundreds of introductory resources that explain conditional probability by reference to Venn diagrams, and this gives you a nice visual illustration of the concept.  You will also be able to find exercises to test your knowledge.  I would recommend this as a starting point for anyone having difficulty understanding the relationship between conditional probability and the AND/OR operations on events.
A: When you say 
$$P(A \text{ and }B)=P(A)\cdot P(B)$$
that is true for independent events. For non independent events, the chain rule for probability says:
$$P(A\text{ and }B) = P(A|B)\cdot P(B)$$
In fact, when $A$ and $B$ are independent, $P(A|B)=P(A)$ and you get the first definition back.
The event $A|B$ makes sense when the probability of $A$ is influenced by the outcome of $B$. Its probability $P(A|B)$ is defined as Tim said before.
