A conditional probability distribution occurs when you have two random variables that are jointly distributed. This means that if we have two variables A and B, knowledge of one of the two variables will give some knowledge of the other variable. Then, the conditional probability distribution is the probability distribution function of one of the two variables given the other variable. For example, the probability of A equal to any 'a' given B = b would be described by the following:
This is the case for discretely distributed values of A and B. If instead, the distribution was continuous, then we would have to use the conditional density function. In which case you would get:
Where the numerator gives the joint density between A = a and B = b, and the denominator is the marginal density.