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A slitless spectrograph is a spectrograph with a field of view (FOV) that is not limited to a slit shape. A typical spectrograph's slit limits its FOV to be very narrow along one dimension, resulting in an image (2-dimensions) that varies with the source over one of its dimensions and produces corresponding spectra over the other dimension. Without the restrictions of the slit, the dispersed incoming light can end up blended, depending upon the size and position of sources within the FOV. Slitless spectrography is the use of a slitless spectrograph.
One use of a slitless spectrograph is to serve as a multi-object spectrograph for a number of point sources that are considerably spread out over an otherwise "blank" field. The spectrograph's output of such a field of stars is a line-segment for each individual star, showing that star's spectrum. Too many stars crowding each other would result in confused output, as would an extended source. This approach was used in the early 20th century for building star catalogs.
Another use is to record rapidly-changing spectral data over extended sources (which might seem to be the worst possible use). Solar phenomena constitute the prime example: there is a need to record such spectral information about active solar flares, coronal mass ejections, etc. Spectral data over a whole field is desired (e.g., from an integral field spectrograph), with frequent snapshots to track the spectra over time, and the limited FOV of a slit limits its effectiveness. A slitless spectrograph produces a confused raw result, but post-processing can sometimes extract useful data, with processing somewhat akin to that for speckle suppression. The fact that the source is particularly bright means short integration times are useful, thus making possible very frequent snapshots of the spectra.