A proof is any rough, unfinished image. Usually with no afterwork, not even cropping, often with a watermark or something, used for selection purposes.
Almost invariably, the final finished image used for output viewing, by whatever means, is a JPEG. It is compressed, but a high quality JPEGs loses no visible data. They are indistinguishable from a TIFF, which in simplest terms is just a visible version of the Raw - you can't actually see a Raw file, it's just a load of digital data.
Compression works by taking common apsects of an image and reducing the sheer volume of data mathematically. For example, if you have an area of absolutely plain tone that occupies say 1m pixels, then in a Raw/TIFF that would be (very simplistically) 1010101010101010101010101010... and so on and on written one million times. The JPEG would just reduce that down to 10x1m, and it would look exactly the same.
The different JPEG quality standards take a different view of exactly what those 'common aspects' are. Taking an area of graduated tone from light to dark, a high level JPEG might reduce that down to 100 different tones, and a low grade JPEG might reduce it down to just 10.
Some images compress more readily that others. For example, if you have a seascape with just plain blue sea, sky and sand of broadly similar tone levels, then a huge file can get knocked down massively even in a high quality JPEG. But if there is lots of detail in a scene, and lots of tones from light to dark, then that image can't be compressed to anything like the same extent.