the joint pmf and joint pdf
The joint distribution is the idea; the joint pmf or pdf is the concrete formula you actually compute with. It is the two-variable cousin of the ordinary pmf (for variables that take separate values) and the ordinary pdf (for variables that vary continuously).
For discrete X and Y, the joint pmf is p(x, y) = P(X = x AND Y = y): a table of probabilities, one cell per pair, all non-negative and summing to 1 over the whole table. To find the probability of any event, add up the cells in it. For continuous X and Y, the joint pdf f(x, y) is a height over the plane; probability is VOLUME under that surface: P((X, Y) is in region A) = the double integral of f(x, y) over A. The whole surface integrates to 1. As with one variable, the density itself is not a probability — it can exceed 1 — only volume over a region is a probability, and any single exact point has probability zero.
From the joint pmf or pdf you compute everything downstream: marginals by summing or integrating out the other variable, conditionals by dividing, expectations of functions g(X, Y) by the law of the unconscious statistician summed or integrated against f, and covariance and correlation. It is the workhorse of the whole multivariate picture.
Suppose f(x, y) = 4xy for 0 < x < 1 and 0 < y < 1 (and 0 elsewhere). It is a valid joint pdf because the double integral over the unit square is 1. Then P(X < 1/2 AND Y < 1/2) = integral from 0 to 1/2 of integral from 0 to 1/2 of 4xy dy dx = (1/4)(1/4) = 1/16.
A joint pdf must be non-negative and integrate to 1; probabilities are volumes under it.
A joint pdf value f(x, y) is a density, not a probability — it can be larger than 1. Only its volume over a region (an integral) is a probability.