The word isomer is derived from the Greek iso meaning equal and meros meaning share. Thus isomers share equally. In chemistry they are defined as molecules that are constituted by the same number and type of elemental atoms but have different properties. In physics they are defined as two different physical forms of the same element, thus having the same atomic number and mass. The reality is that even amongst equals there is a primum inter pares and some as the cynic quipped are more equal than others. In chemistry the historically monumental isomers are Ammonium Cyanate written as NH4CNO and Urea written as H-N-H
The first is an inorganic compound and the second is an organic compound. It was believed that organic compounds were products of living organism. It was the German Chemist Wohler who effected the conversion of Ammonium Cyanate to Urea by heat and broke the barrier. In physical terms the critical isomers are graphite and diamond, both forms of carbon in amorphous and crystalline form and having different hardness, appearance, specific gravity a melting points and human value. One is mightier than the sword and the other is a woman's best friend if you believe advertising hype.
Isomers are much more common in organic chemistry with its branching structure. An interesting pair is dimethyl ether CH3-O-CH3, which is a highly flammable gas at room temperature and an aerosol propellant and ethyl alcohol CH3-CH2OH which is a solvent and a depressant but used as a drink for celebration and as a social lubricant. When the connectivity of atoms differs as in the above example the two are called constitutional isomers.
There is a special class of organic isomers which are identical in structure except that they are mirror images of each other around a single asymmetric atom fulcrum. These are stereo-isomers of a special type called enantiomers and have fascinating biological significance. It was Louis Pasteur who discovered that tartaric acid in the dextrorotatory (rotates polarized light to the right) form was present in ripening grapes. He found that certain chemical reactions racemized the tartaric acid. This solution was an equal mixture of the dextro and levo rotatory forms which crystallized in identical but mirror image shapes and forms (like our hands which are identical but cannot be superimposed right on left while facing the same way). He also discovered that a solution of racemized tartaric salts when seeded with yeast left a residue of only the L tartrates by selectively metabolizing the D variety. Thus the selective preference for specific varieties by biological organisms was discovered. The selective preference of utilizing D glucose by living organisms was a related discovery. When there are two asymmetrical atoms in a single molecule we get diastereoisomers.
Far more interesting is the fact that all living organisms have almost all their proteins made up of L amino-acids. Is this an accidental artifact of history because life began with the L amino-acids or is there another plausible explanation? D amino-acids like serine often act as antagonists (hindrances like inserting a wrong key that gets jammed and ruins the lock without opening it) at certain neuronal receptors like the N methyl D aspartate one. Spontaneous isomerization to D form amino-acids within certain lens proteins (which are linked chains of amino-acids) may be one of the etiologies of cataracts.
Certain drugs like thalidomide have D and L enantiomers. Only one is teratogenic but since the human body interconverts both forms treatment with one enantiomer is a moot point. Another biologically interesting fact is that our sense of smell has evolved to distinguish nutritionally relevant fruit smells mediated by the enantiomers of limonene. R limonene smells of oranges and L limonene of lemons. Tests with other enantiomers not having evolutionarily nutritious value often fail in our ability to distinguish the two mirror image compounds.
Finally due to the rigidity of a double bond between two carbon atoms two different ligands (attachments) maybe on the same side or on diametrically opposite sides. This variety of stereoisomers are called cis and trans. Again the living organism treats the two differently and its metabolic constraints maybe used therapeutically or for better health. Cisplatin is a cis isomer used to treat certain cancers. The trans variety is not as effective. Finally the trans variety of fats and fatty acids are far more unhealthy and lead to greater artherogenesis while the cis variety are metabolized with less dire consequences. The selective synthesis of enantiomers by biological systems of organisms was finally mimicked by the pioneering work of Knowles, Noyori and Sharpless for which they received the 2001 Nobel Prize in chemistry.
To end on my favorite political note, nature is chiral (handed) and leans preferentially to the left! Another way to put it is that if the left is right then the right must be wrong and the center directionlessly racemic!