Generally, when the "proof" of an alcohol solution is used, one is dealing with a solution of mainly alcohol (that's ethyl or grain alcohol) in water. The value of the "proof" is exactly twice the percentage of alcohol in the solution. Pure alcohol is "200 proof." A solution that is 43% alcohol would be "86 proof."
The percentage of alcohol in a water solution (and thus the "proof") is usually determined using a hydrometer, a device that measures the density (or the specific gravity to be more precise) of the solution. As the percentage of alcohol in a solution increases, the density decreases because alcohol has a lower density that water. The hydrometer is a floating device that has a long stem that sticks out of the solution. As the density of the solution decreases the hydrometer sinks further into the liquid. The amount of the stem that sticks out of the solution is a measure of the percentage of alcohol or the proof - the more of the stem that sticks out of the solution, the lower the "proof".
It is interesting how the word "proof" became used for the amount of alcohol. Back in the old days (pretty far back) when people were making and selling alcoholic beverages, there was a concern for just how much alcohol was in them. Those which had more alcohol were prized. A qualitative test was developed. Some of the alcoholic beverage was poured onto some black powder and an attempt was made to ignite the powder. If the liquid had too much water in it, the powder would not ignite. If the powder did ignite, this was "proof" that it had a high enough amount of alcohol in it.