Camera sensors have many different sizes. A full frame sensor is 36x24mm in size, and a camera sensor that is smaller is regarded with a certain degree of ‘crop factor’.
The degrees currently in general use are shown below.
Another factor must be taken into consideration. If you used a full frame camera to photograph a landscape with a 50mm lens, the scene would appear the same size as it is in reality (50mm is considered 1x zoom). On the other hand, if you used a 1.5 CF camera, the lenses focal length would be equivalent to 75mm. A 1.6x CF sensor with 50mm lens has the equivalent to 80mm, but be warned, this can be confusing. The focal length is not actually altered, simply the angle of view.
While it seems that discarding information in this way is not only wasteful, but also annoying, it does have its advantages. For example, the image is likely to have more overall sharpness. This is enhanced with low quality lenses, because the edge information is usually the worst quality.
A larger sensor has a shallower depth of field. This means that to achieve the same depth of field, you need a narrower aperture. A shallower aperture is often prefered for portraits and macro..This is why it is easier for DSLRs to produce blurred backgrounds.
Diffraction ”Airy disk” will take over at narrower apertures, when it becomes larger than the circle of confusion and this makes the images softer, but larger sensors can use smaller apertures before this begins to happen. Diffraction limits are adressed here.
Larger sensor generally have larger pixels, and this can create higher dynamic range, and even improved noise reduction.
As sensor size doubles, so does the price. There are also other considerations, including there are less fast lenses available for cropped sensors. The same applies to wide angle or superzoom one.
Finally, tilt ans shift lenses can alter the actual appearance of depth of field.