A probability density function (PDF) is a function on a continuous random variable that gives the relative likelihood of the variable taking the given value. If a function shows the relative density at each possible value of the random variable, but the sum of the probabilities does not equal one, then it can be referred to as an unnormalized PDF and the equivalent normalized PDF (NPDF or N-PDF) can be derived by taking the unnormalized PDF and incorporating an (additional) division by the area under the curve.
A continuous random variable is a value that can have any value over an interval. For example, if you have observed the widgets a factory produces randomly vary in length over an inch, but always between 10 and 11 inches, then you can model the length as a random variable, and perhaps discover a function (a PDF) over the interval of real numbers from 10 to 11 that describes the probability of each corresponding length in inches.
Similarly, a probability mass function (PMF) is like a PDF except that rather than mapping into a continuous random variable, it maps into a discrete random variable, i.e., a finite number of values.