An imaging spectrometer is a spectrometer that collects spectral data over a two-dimensional area, i.e., for each part of the image, information about the intensity for each wavelength is determined. This produces a "cube" of data (three dimensions), two being spatial (the image) and one being the spectral data at the specified point.
There are various types: a common type for astronomy is the integral field spectrograph which incorporates a device called the integral field unit to arrange pieces of the two dimensional image along a line, i.e., one dimension. Common spectrometers use a slit to image and measure the spectrum across a line selected from the telescope's two-dimensional image, so the resulting graph (photo or electronically stored image) displays spectral data in one direction (the spectral direction or dispersion direction) and spatial information over just one dimension in the other (the spatial direction). By the rearranging the image so as to place image-portions across a two-dimensional area so they are in a line, the same spectrograph can capture spectrums of the whole image at a lower spatial resolution. Lenses, prisms, and/or fiber can be used to accomplish the rearrangement. This constitutes a time-saver compared to taking a series of snapshots with a "normal" spectrometer, shifting the slit each time to capture the second spatial dimension.
Another type of imaging spectrometer is an imaging Fourier transform spectrometer.
Be warned that the term imaging spectrograph has also been used for a spectrograph solely because it has a mode of operating as an ordinary camera, i.e., offers an option of bypassing the prism or grating, so as to use its camera for (non-spectroscopy) imaging. Such an instrument may be described as an "imaging spectrograph" (e.g., the Hubble Space Telescope's STIS) even though in spectrograph-mode, it views only the typical slit's single dimension.