Optical depth (τ) is a measure of opaqueness, i.e., the degree of absorption of light passing through a medium. It is the integral of the product of the material's opacity and density over the path of the light. It is equivalent to the number of mean free paths of a photon as light passes through the medium (if they were laid end-to-end on a line). While opacity is a quality of the material, optical depth measures the effects of light passing through a specific length of material, accounting for variations in the opacity and density along the length. But like opacity, it varies by wavelength.
In dealing with the atmosphere of stars, optical depth is generally assumed to mean the optical depth from above, specifically, starting with 0 at the top of the atmosphere. The optical depth specifically measured perpendicular to this surface is known as the vertical optical depth, a quantity that is useful in some analyses.
The Rosseland optical depth is the optical depth calculated using the Rosseland mean opacity, i.e., a weighted average (across frequency) opacity.
Intensity falls exponentially by the optical depth, so if it is below 1, intensity falls slowly and the material is called optically thin (i.e., less than the mean free path of a photon and a given photon is expected to escape) whereas if the optical depth is much greater than 1, intensity falls quickly and the material is called optically thick (i.e., more than the mean free path of a photon).