Two random variables, x and y are independent if the occurance of one does not give any information on the likelihood of the other event occuring. In other words, their probability distribution can be expressed as a product of two factors, one only involving x, and one only involving y:

Two random variables, x and y are conditionally independent if, given knowledge of the occurance of Z, knowledge of the occurance of X provides no information on the likeihood of the occurance of Y. This can be expressed as: