Monoamine is a term used in biochemistry, meaning literally "single amine", but actually used to mean a chemical that is derived from a single amino acid.
The basic biochemistry of amino acids is that they are used to make different proteins, either for structural purposes or for enzymes. However, there are other roles needed by the cellular machinery, so sometimes a single amino acid is repurposed and modified structurally into a monoamine. Examples of endogenous monoamines include the neurotransmitters serotonin and melatonin (derived from tryptophan) and dopamine and epinephrine (derived from tyrosine). Monoamines make good neurotransmitters because their small size means they diffuse rapidly.
The most important pharmacological fact about monoamines is that they pass through the digestive system intact, and also can be absorbed through mucous membranes. A protein is too large to pass through mucous membranes, and when it goes through the digestive system, will be cleaved into its component amino acids. A monoamine, on the other hand, will be absorbed into the body as soon as it is ingested, passing into capillaries in the mouth and cheeks, and will pass through the stomach unchanged and be taken up in great quantity through the small intestine. This is why monoamines, or monoamine-like substances, such as cocaine or LSD, can be ingested and have an effect, while someone could have a large pile of oxytocin, endorphines and other psychoactive substanced that are proteins, and could get no effect from ingesting them or applying them to a mucous membrane, because they are too large to go through a capillary, and would be broken up in the stomach.
Monoamines are deactivated by monoamine oxidase, a protein that plays an important role in brain chemistry. A group of drugs called monoamine oxidase inhibitors blocks this breakdown, causing monoamines to stay active in the brain, which can be a very dangerous thing.
As a note on terminology, "monoamine" properly belongs to substances that are derived from a single amino acid, but much of the pharmacology can apply to other substances. Caffeine, for example, is not derived from an amino acid, but is about the same size and shares the property of being absorbed through mucous membranes and passing through the digestive system intact. So it would probably not be wildly inaccurate to refer to caffeine as a monoamine.