What are the types of molecules that exhibit fluorescence?

In general, molecules that fluoresce are conjugated systems, and thus are linked together allowing electrons to move between each molecule. Fluorescence occurs when atoms or molecules relax through vibrational relaxation to their ground state after being excited by UV light. As the molecules relax, it releases energy as visible light and heat. Excitation and emission wavelengths are distinct to each molecule or atom that exhibits fluorescence. 

Fluorophore molecules are classified into four categories: small organic compounds, proteins and peptides, multi-component systems, and oligomers and polymers. Some examples of non-protein organic fluorophores are xanthene derivatives (fluorescein, rhodamine), coumarin derivatives, and cyanine derivatives (cyanine, merocyanine). Small-molecule organic fluorophores are typically used in applications such as single-molecule and super-resolution microscopy. They are also the smallest of known extrinsic fluorophores (only a hundredth the size of fluorescent proteins and quantum dots). It is important to note that fluorophores can either act on their own, or be a part of a functional system.