When light from a star or another astronomical object enters the Earth's atmosphere, the different temperature layers and different wind speeds distort and move the image in various ways. The distortion changes at a high rate, typically more frequently than 100 times a second. In a typical astronomical image with an exposure time of seconds or even minutes, the different distortions average out as a filled circle. The diameter of this circle is measured as the seeing disk.
It follows from this definition that seeing is always a variable quantity, different from place to place, from night to night and even variable on a scale of minutes. Astronomers often talk about "good" nights with a low average seeing, and "bad" nights where the seeing was so high that all observations were worthless.
Seeing is usually measured in arcseconds, abbreviated with the symbol ("). A 1.0" seeing is a good one for average astronomical sites. The seeing of an urban environment is usually much worse.
Seeing is one of the biggest problem for Earth-based astronomy: while the big telescopes have theoretically milli-arcsecond resolution, the real image will never be better than the average seeing during the observation. This can easily mean a factor of 100 between the potential and practical resolution.
The first answer to this problem was NASA's Hubble Space Telescope: working outside the atmosphere, this telescope does not have any seeing problems, and always works at maximum resolution power. The difference is so great that, even being quite small (only a 2.4 meters mirror), it surpassed all Earth-based telescopes for many years, and it is still widely regarded as the best one.
Currently, many telescopes are developing adaptive optics systems that partially solve the seeing problem, but none of the systems so far built or designed completely removes the atmosphere effect.