Definition
A probability distribution describes how probability is allocated across the possible values of a random variable. For a discrete random variable, it is a list of values paired with their probabilities (a probability mass function). For a continuous random variable, it is a density function — areas under the curve give probabilities for ranges of values.
The distribution is the complete probabilistic specification of the random variable; everything else (mean, variance, percentiles, generating functions) is derived from it.
Why it matters
How it works
Each distribution has a name, a set of parameters that fix its specific shape, and formulas for its key properties. The normal distribution has parameters mean and standard deviation; the Poisson has rate; the binomial has number of trials and per-trial success probability.
To use a distribution, you identify the random variable, estimate or specify its parameters, and then read off the probability of any event by integration (continuous) or summation (discrete). Modern statistical software lets you do this with a single function call, but the underlying machinery — densities, masses, CDFs — is the working language of every applied probability problem.