Astronomical seeing is the blurring of objects seen through the atmosphere. When using a ground-based telescope, even a long exposure will result in point and line details being turned into finite blobs and "bars", due to continuous changes in the optical characteristics of the atmosphere.
A seeing limited telescope's limits of angular resolution are due to this phenomena. In contrast, a telescope in space or a ground-based telescope that can cancel these effects is limited by diffraction and is termed diffraction limited.
Atmospheric effects are caused by density variations in the atmosphere over time and space. These effects can be reduced by locating telescopes at high altitude, by favorable weather conditions, by large telescopes, and by adaptive optics. The effects vary over time, some nights being better than others (i.e., calm nights), and the seeing may change by the minute.
A seeing number is a measure of seeing, an expression of how far from "true" a point might appear on the focal plane, or as observations often take time, and the atmospheric effects change continuously, how big the spot is on the focal plane from the point (e.g., a distant star) due to the atmospheric turbulence. Since the aperture of the telescope already limits its resolution by turning the point into an Airy disk, the seeing number may be added to that resolution to give some hint as to the resolution you are getting. The seeing number is typically the full width at half maximum, i.e., given an exposure time much longer than the changes in seeing due to turbulence, the point will be seen at different points in time around a circle in the focal plane, mostly in the center, and the further from that center, the less often. The FWHM seeing number is half the diameter of the circle that includes the most extreme distortion, and is expressed in arcseconds, as is a telescope/instrument's resolution due to Airy disks.
Other measures are r0 (Fried parameter) and t0 (Greenwood time constant). The former is an aperture size sufficiently small that seeing distortions would be minor, and the later is a time interval estimating how long an exposure could be before seeing distortions are likely to be significant. Aside from describing seeing, they are also basic parameters in the design of AO systems.