Optics:
The far field is quite intuitively the optical field far from where some interesting thing is happening where things don't change much anymore. Most often used in the context of fourier optics to describe where Fraunhofer diffraction is a valid approximation of fresnel diffraction. Near the image plane (the near-field), light diffracted through an aperture of some kind will undergo some interesting changes until finally settling down to a pattern that just gets larger as it propagates further. The point where it is settled down and beyond is called the far-field. There is a back of the envelope formula for computing the far field which is given by:
2*D2/lambda
Where D is the size of the smallest feature of the aperature and lambda is the wavelength of the light used.
When you go to a football/soccer/baseball game at night, the stadium you may have noticed that the lights sometimes have a groovy star shape. This is the far-field diffraction of the lights through the metal wire shield surrounding the bulbs.