Note: The following answer was originally written in answer to another question that, while very similar to this one, was concerned specifically with the differences between sensor sizes when shooting in low light situations.
Will 1 Inch sensor give same exposure at same aperture and ISO settings compared to APS-C sensor?Will 1 Inch sensor give same exposure at same aperture and ISO settings compared to APS-C sensor?
Exposure is a measure of the field densityfield density of light. The means it is an expression of how much light is captured per unit of area.
If you have the same ISO, f-number, and shutter time you will get the same exposuresame exposure. There can be minor differences due to the inaccuracies of different cameras with regard to actual ISO, Shutter time, and aperture as well as the varying amount of light that is lost as it travels through various lenses. But for creative photography purposes anything within about 1/6 to 1/3 stop is viewed as close enoughclose enough.
What you lose with a smaller sensor, especially when shooting in very low light conditions is the total amount of light collected. When the field density of light is the same, the amount of light falling on each square millimeter is the same, but the sensor that is four times as large in terms of area collects four times as many photons spread over four times the area. Assuming the angle of view is the same with both cameras due to different focal length lenses, the brightness of each mm² will be the same but the larger sensor produces a larger image. This is significant when we enlarge the image from the size it is on the sensor to the size with which we wish to display it.
If the images from both sensors are enlarged to the same display size, the image from the larger sensor requires less enlargement than the image from the smaller sensor. When images are enlarged from the size they are projected onto the sensor everything gets enlarged: the image from the light that was projected onto the sensor and recorded, the noise generated by the camera, the noise created by the random nature of light, blur due to motion and focusing/DOF issues, and any optical imperfections due to the lens.
So in the end what a larger sensor gives you is the ability to enlarge less to get to the same display size which means all of the imperfections in the photo are not as magnified as they would be with a smaller sensor.
For some situations, though, there are techniquestechniques that will allow the performance of both the smaller and larger sensors to be improved. Shooting at lower ISO for a longer exposure, for example, will reduce the influence of photon shot noise. Of course that might require a tripod or other means of stabilizing the camera. Using dark frame subtraction can reduce the influence of constant read noise produced by the camera. Stacking multiple images of the same scene will reduce the random noise in each frame. Stacking almost certainly requires a tripod. But any improvementsany improvements you make using the smaller sensor can also be made using the larger sensor. Thus the larger sensorlarger sensor will always maintain its light collecting advantagelight collecting advantage when both are based on the same technology.