When hydrogen from hydrogen gas (H2) or another molecule combines with another chemical compound, the process is called "Hydrogenation".
The compound becomes saturated or reduced. As hydrogen donates electrons, the other compound gets reduced. Hydrogen gets oxidized. This process is used in food industry to convert unsaturated oil (fats) into saturated form. Like butter, vegetable oil etc.
Hydrogen donates the electron and becomes a positive ion and forms single bonds with the other molecule.
In inorganic chemistry, Hydrogen combines with halogens to form halides like HCl (gas or liquid). With Oxygen group elements it forms water, hydrogen sulphide gas etc. With metals it forms hydrides.
Hydrogen reacts with most elements. It forms the most important compounds called acids and bases in the form of hydroxides.