A first-order process is a process that has a rate proportional to the amount of the reactant involved in this process.

For example, if the dg elimination rate is proportional to the amount of the drug in the body, this means that this drug is eliminated by a first-order process. The proportionality constant is the first-order elimination rate constant. i.e.

Elimination rate = elimination rate constant x drug amount

The rate of drug elimination decreases with time because the amount of the drug remaining decreases with time.