In astronomy, background commonly refers to the incoming light from an apparently empty part of the night sky.
Even if no visible astronomical objects are present in given part of the sky, there always is some low luminosity present, due mostly to light diffusion from the atmosphere (diffusion of both incoming light from nearby sources, and of man-made Earth sources like cities).
In infrared astronomy, the problem can be much worse: due to the longer wavelengths involved, the sky and the telescope themselves are a source of light.
[1] To work around this problem, infrared telescopes often use a technique called chopping, where a mirror rapidly oscillates between the object of interest and the nearby, empty sky.
Even if we able to minimize the effect of instrumental and atmospheric components (e.g. using a spacecraft), there are still several astrophysical components contributing to the sky background: these could be sets of point sources like faint asteroids, Galactic stars and far away galaxies, as well as diffuse sources like dust in the Solar System, in the Milky Way, and in the intergalactic space.