The PDF, short for probability density function, is a derivative of the cumulative density function, or CDF, according to the Wolfram Demonstrations Project on connecting the CDF and PDF. Both functions are used in probability theory and in statistics.
The PDF is a function that describes the relative likelihood for a random variable to take on a certain value. The CDF, also known as just the distribution function or DF, describes the probability that a real-valued random variable with a given probability distribution has a value less than or equal to X, according to Wikipedia.
These two functions work together to answer questions put forth by the other. For example, according to Wolfram, the PDF answers the question of "How much of the distribution of a random variable is found between observation values?" On a graph, the CDF is more helpful because students can use it to estimate the probability of a certain observation within that range.
Wikipedia explains that probability theory is a branch of mathematics that deals with probability, or the analysis of random phenomena. Central objects of this theory include random variables, stochastic processes and events. For example, if a person flips a coin or tosses a die as a random event, then the repetition of that random event eventually begins to create certain patterns; and those patterns can be studied and later predicted.