Fourier space (or frequency domain or Fourier domain) is a space into which the Fourier transform maps a function, consisting of the amplitude and phase of the sine function at various frequencies that sum to produce the same shape. For a function that is periodic or is over a finite domain, the frequencies generally chosen are with periods 1/2, 1/3, 1/4, (and so forth) of the repeating period (or domain extent). There are ways to apply the Fourier transform to a function of one or more dimensions, two dimensions being commonly used for processing images.
"Normal" space for, say, a two dimensional image could be X and Y, with a function consisting of the amplitude at each such point. Fourier space is two corresponding frequencies (periods), with the transformed function mapping the frequencies them into an amplitude and a phase of a sine wave (a complex number) such that if all were summed over the fourier space, the same surface is produced.
Operations such as truncating (removing precision) affect an image differently than would applying it to the normal space, offering more means of manipulation and analysis.