If you put neat alcohol onto your skin and leave it there for any length of time it will dry your skin out. This is something that is well known. The exact details of how it works aren’t that well studied, but basically it disrupts your skin’s barrier function. Consequently it is not a good idea to put a lot of alcohol into products intended for the skin. But alcohol is a useful solvent and it often ends up on ingredient lists because there is a small amount in the formulation somewhere. A small amount of alcohol is quite harmless.
To chemists the term alcohol is a generic one for a whole family of chemicals. The one that gets you drunk is ethanol. The intention of ingredient lists on cosmetics is to inform people with specific allergies which product they are to avoid, so the name on the ingredient list is alcohol.
There are two main forms of alcohol that end up in products – pure alcohol and denatured alcohol. Denatured simply means that a small quantity of something foul tasting has been included in it to make it difficult to drink. There are very few cosmetic products that contain enough alcohol to be a practical way of getting inebriated and even fewer where doing so would be remotely cost effective. The denaturing is intended to stop the people who make the cosmetics misusing the raw materials rather than end users.
But as I said, ethanol is far from the only member of the alcohol family and other alcohols turn up on ingredient lists. Materials like cetyl alcohol are fatty alcohols which are generally waxy solids at room temperature and which have no intoxicating properties at all. They are also not harmful to your skin either, so there is no need to avoid them.
I have been asked once by a Muslim woman if she should avoid any cosmetic product that contained any kind of alcohol. The Koran predates organic chemistry by several centuries so there is no way that it could ban the use of any particular kind of chemical compound, so alcohols other than ethanol are in the clear. I would have thought that the intention behind the ban was to prevent the faithful clouding their minds so I don’t think there can be any harm in using actual alcohol in cosmetic products. It is well below the level it could possibly affect your judgement. But if you choose to do so, you need to avoid both pure ethanol, which is listed as the single word alcohol and denatured ethanol which is listed as alcohol denat. If the word alcohol is preceded by anything on the list, typically things like benzyl alcohol and stearyl alcohol then these are not ethanol and there is no problem.
The name alcohol is of arabic origin and pure alcohol was probably first produced by Arab chemists. The great Persian philosopher Avicenna has left some of the earliest accounts of distillation. This activity in the Middle East in the tenth century was the foundation of the modern herbal extracts industry which continues to come up with new compounds, and alcohol is both a product of that industry and a solvent that gets used to extract other things.