"Medium Format" is a term used in photography to describe a category of cameras, and the film they use, which is larger than 35mm but smaller than 4 by 5 inches, which is the starting size of large format photography.
The common denominator for medium format size 120 roll film is that one dimension (we'll call it the width) of the film is 6cm. The other dimension (we'll call it the length of the exposure) depends on the type of medium format camera you are using...the most common sizes are 4.5cm, 6cm, 7cm, 8cm, and 9cm. So, when you see a camera described as a "6x6," you know it is a square format camera (like a classic Rolleiflex twin lens reflex). There are also panoramic film backs on some specialized cameras that make images 6x12cm and 6x17cm.
The following graphic shows the relative size of the most common medium format size exposures, compared to the size of 35mm negative.
The designation of medium format roll film as "120" is a bit confusing. The number is merely a designation for the type of film, and has no relationship whatsoever to the size of the film. The numeric designations are simply sequential numbers assigned to new film formats that came along over the years. These numbers were introduced by Kodak, which began allocating the number series in 1913. Prior to that, because there were no standard and because film was camera-specific, films were identified by the name of the cameras in which they were used.
Film designated "220," is the same format as 120 except that the film itself is twice as long, doubling the number of exposures per roll. 120-type film has hung on and become the de-facto standard for medium format film photography, while production of 220 film has virtually ceased.
The advantages of medium format film over 35mm is mostly obvious: a much larger sized negative (high "resolution"), results in the ability to make much larger, sharper prints. But it is more than that. There is a certain quality of the image itself that stands out; there is something about the perspective and the "feel" of the images that just seems more natural and pleasant.
While the term "medium format" was born in film photography, it has been adopted in the digital world to define digital cameras with image sensors larger than 35mm. In digital SLR terminology, "full-frame" means that a sensor is the full size of the classic 35mm negative...3.6x2.4cm. These are relatively expensive cameras, so if you have done the math and surmise that medium format digital sensors must be really expensive (since they can be four times larger than a full-frame 35mm sensor), you would be correct!
In a wonderful stroke of unanticipated genius, the original modular design of the Hasselblad film cameras in the 1950-1980 era enabled it to be later adapted to a hybrid analog/digital configuration. In the 1990s, some innovative digital technologists looked at the Hasselblad camera systems with their removable film backs and realized they could create a digital back of the same profile to replace the film backs. Hence, photographers were then able to enjoy the beauty, optical quality, and ruggedness of the Hasselblad, at the same time enjoying the convenience and power of digital imaging.