Essentially, in conditional probability, when it says "given A" it is telling you that we know for sure that event A has occurred, so we are now only interested in outcomes that belong to A.  That becomes the "whole".  P(B|A) wants the fraction of that "whole" that also belongs to B.

For example, if you look at my question 18 in the probability handout from my Basic Stats 1 Probability Lesson above, I could add a part (d) that asks, "What is the probability someone is a basketball fan if they are a hockey fan?"  Any probability question that asks, what is the probability of B if event A has occurred, you are doing conditional probability.

We want P(B|H).  I first look through my Venn diagram and find all the bits that belong to H, since we know for sure the person is a hockey fan. There are four bits in the H circle so I add those bits up: 33 + 31 + 8 + 5 = 77%.  Now, I gather all the bits in that H circle that represent people who are also basketball fans.  There are two bits: 8 + 5 = 13%.  Thus, the probability a person is a basketball fan if they are a hockey fan is 13%/77% or .13/.77 = .1688.

Here is a couple of extra conditional probability questions I have added to question 4 in my Probability Lesson handout above: