A monomer is a molecule that can bind to (i.e. form bonds with) similar molecules and form chains, which are known as polymers. There can be a very large number of monomers (even thousands) in a polymer. The process by which monomers form chains is known as polymerization.
Some of the common molecules, such as carbohydrates, proteins, lipids and nucleic acids are polymers. Carbohydrates consist of monomers called monosaccharides, proteins contain amino acids, lipids contain glycerol & fatty acids and nucleic acids contain the monomers called nucleotides. Similarly, a commonly found monomer, such as glucose, polymerizes to form cellulose, starch and glycogen.
Generally, monomers are organic molecules that form organic polymers (such as those mentioned above). Also remember that polymer properties vary as the number of constituent monomers increases.