A function that assigns a probability to each possible value in the case of a discrete variable or range of values in the case of a continuous variable. Such a variable is called a random variable.
Each probability must obey the usual rule that it range between 0 and 1. Furthermore, in the case of a discrete random variable, all probabilities must sum to 1. Similarly, for a continuous variable, the integral from minus infinity to infinity must be 1. All this says is that it is certain that the variable will have exactly one of the possible values.
Certain probability functions are common in nature, hence they've been studied extensively and the math for manipulating them is well known. The uniform distribution assigns equal likelihood to each value of a discrete variable; when graphed, it appears flat. The binomial distribution is related to combinations of random two-valued events. For example, if you flip a coin ten times, the number of heads will range from zero to ten, with an average of five. If you repeat this process, you'll see that five heads is more common than four heads, four is more common than three, etc., and if you graph the frequency of each of these counts, you'll see something similar to a bell-shaped curve (although it won't be smooth because you're only plotting eleven points). This looks much like the well known normal or Gaussian distribution that's applied to continuous variables.
Probability distributions can usually be expressed as a type with one or more parameters that define the specific distribution. For a normal distribution, we typically use the mean (average) and the variance or standard deviation, which measures the "spread" around the mean.
In real life, data often appear as combinations of multiple distributions. Teaching assistants often notice that test scores in university classes are typically bimodal, that is, having two distinct "bumps" in the curve. These correspond to the means of two different distributions added together: the students who get it and the students who don't get it.