Mineral oil is a derivative of petroleum. It is scientifically generally accepted that petroleum is generated from heating fossilised organic material that decomposed over the years in the absence of oxygen, like for example deep in the sea, and is possibly the product of dead prehistoric zooplankton, algae and other organisms. As a result, petroleum and by extend mineral oil are considered to be natural materials.
As you can imagine, heating and decomposition of dead organisms will result in a mixture of compounds, mineral oil being one of them, and therefore, petroleum has to undergo several purification processes in order to extract the mineral oil or other compounds of interest.  As a result, mineral oil can be characterised as a “clean part” of petroleum. Despite the very simple chemical build of this molecule, a hydrocarbon molecule made of only Carbon and Hydrogen elements, it is very commonly used in cosmetics, food and pharmaceuticals.  Mineral oil is usually a combination of two families of hydrocarbon compounds called the parraffinics and the naphthenics. The difference between them is in the actual structure of the molecule, ie. how the Carbon and Hydrogen atoms are placed. But while we don’t need to bother ourselves with further chemical details, it is important to understand that mineral oil is found in commercial products as a mixture of compounds, belonging to the two families of compounds just mentioned above, and the ratio between them is what gives mineral oil different properties for different product applications.
Mineral oil is similar in function to vegetable oils but can also be different depending on its exact composition. However, it is speculated that the use of mineral oil is often preferred because of its low cost.  Although both mineral oil and vegetable oils are natural products, mineral oil does not fall under the “natural” category in the cosmetic industry, as it is a mixture of compounds that is derived from heavy processing. On the other hand, mineral oil is made up of Carbon and Hydrogen atoms only, meaning that it is highly inert and unlikely to interact with other things making it a very good candidate for cosmetics as it can also provide some level of skin moisturisation. 
Scientific research on the moisturisation properties of mineral oil has been very interesting. Some studies concluded that mineral oil does not possess any moisturisation abilities  which in a way is true. However, other research studies give evidence on mineral oil being a moisturiser but not in the traditional way. Mineral oil acts as a barrier, a shield if you like, for preventing water loss. As a result, while mineral oil is not a moisturising cocktail itself, it can indirectly improve moisturisation and softness levels of the skin.  Although, don’t expect anything as amazing as for example, hyaluronic acid – the king of moisturisation. (see post “Hyaluronic acid – the fountain of youth?“)
Mineral oil is said to be safe not only because of its chemical inertness (doesn’t react with other things) but also due to the fact that it does not penetrate the skin deeply with only about 1-2% penetrating deeper skin layers and sometimes entering the bloodstream.  If the thought of what you put on your skin entering your bloodstream freaks you out, bear in mind that a lot of skincare products can do that, even more reason to invest in quality skincare products. You wouldn’t put anything in your mouth so why would you put anything on your skin?
Unfortunately, several studies have shown that mineral oil is not comedogenic.  This means that mineral oil does not prevent any acne breakouts caused by the use of cosmetics. Additionally, mineral oil has been found to increase light penetration into the skin slightly labelling it by some as a photocarcinogen. The evidence on this however, is limited and there is no reported tumor development linked directly to mineral oil.
In a recent study by P. D. Forbes it was concluded that mineral oil can indeed increase the skin’s light sensitivity by 5-7.6%.  However, while that number might sound worrying it is actually fairly low when compared to the effect of other everyday activities on our skin. For example, using exfoliating treatments on skin is reported to increase the skin’s light sensitivity by 12-13.2%.  Perhaps don’t exfoliate before going to the beach?
Oil induced skin sensitisation by light is actually still debated and is a slightly controversial topic as there are also a couple of research studies that show that mineral oil, especially when in combination with other compounds, had no effect in the skin’s light defences. However, the most general scientific consent at the moment is that mineral oil does indeed increase the skin’s sensitivity but the amount is so small that long-term usage effects are not expected to be serious. For information on how to keep your skin protected by light see post “All you need to know about sunscreens and SPF”.
Although it might sound a little fancy, mineral oil is actually present in a lot of standard and cheap products that you most likely have used at some point already. Perhaps the best example is in baby oil which is basically made of mineral oil and fragrance. Mineral oil is also used to ease infant rashes, mild eczema, bath oils and more.  It is approved by the FDA (US Food and Drug Administration) for medicinal applications in eyes, mouth and topical skin applications.
Other than the cheap and abundant nature of mineral oil, it is speculated that the cosmetic industry has been interested in this ingredient due to the “organic” and “natural” themes that have been trending and growing the past couple of years.
Mineral oil is a colourless liquid that has no taste or smell and is not soluble in water. If you are interested in finding out if your cosmetics include mineral oil search for: mineral oil, light mineral oil, liquid paraffin, liquid petrolatum, mineral oil mist, paraffin oil, paraffinium liquidum, petrolatum liquid, petroleum oil, white mineral oil and white oil. 
Overall, I can’t conclude that mineral oil is especially worth seeking out. There are other compounds that will give you much better hydration, such as “hyaluronic acid“, more long term softness and elasticity and contribute to better skin health. Claims that mineral oil cannot really provide.