The limiting magnitude denotes the faintest object that can be observed. This depends on the aperture and the used magnification. The limiting magnitude is different for visual observations and imaging.
The faintest possible star that can be seen is directly related to the mirror diameter. An approximation for this is given in the table below. (Note that these values vary from observer to observer, and from place to place)
The above table is only valid for pinpoint light sources. For extended objects the properties of the eye/brain combination become more complex, but the same principle as above is valid, i.e. larger diameter = better detectability. See "Visual Astronomy; Optimum Detection Magnification" by Mel Bartels.
The limiting magnitude increases with higher magnification for pinpoint objects. The reason for this is that the object itself will not appear larger despite the higher magnification. It's light remains concentrated in a point. The amount of light is determined by the aperture and is independent of the magnification. The size of the area surrounding a pinpoint object (the "true" FOV) depends on the magnification. The "apparent" FOV is independent of the magnification. Since the "surface" brightness of the background is constant, the amount of background light decreases with higher magnifications.
The upshot is that for higher magnifications you receive the same amount of light from the (pinpoint) object, but the brightness of the surrounding area gets lower. The contrast is therefore higher for higher magnifications.
The above is of course purely theoretical, and assumes that the pinpoint object stay's a pinpoint. Unfortunately this is not true. The infamous airy disk spoils our praxis. As long as the magnification is so "low" that one cannot discern the airy disk, the above holds true. However as soon as you start seeing the airy disk, you have reached the case where the 'pinpoint' object is no longer a pinpoint. And you should refer to the theory of optimum detection magnification.
For photography the situation is a little bit different, since the film is able to integrate the amount of light it receives over a certain time. The table below gives an indication of how the exposure time and mirror diameter influence the detectable magnitude. Apart from the fact that longer exposures are more difficult, there is an upper limit as to what the film can handle.
CCD's also integrate the amount of light, but at low light levels their detection curve is linear. They suffer from thermal noise which increases with temperature. Unfortunately I don't have any data on the limiting magnitude.