A Duchenne smile is the technical name for a genuine smile -- one that involves the muscles around the eyes, and not just the mouth. Of course, it's a little more complicated than this, but if you start to watch smiles, you will soon be able to spot which are actually indicating engagement and happiness, and which are just polite smiles.
Duchenne de Boulogne was a French neurologist who studied the muscles involved in facial expression. He discovered that smiles that involved the orbicularis oculi muscle appeared significantly more genuine than those that do not. This muscle raises the cheeks directly below the eyes and causes crow's feet at the corners of the eyes,
Sometimes writers will refer to non-Duchenne smiles as 'fake smiles', which is not exactly correct. Both Duchenne smiles and run-of-the-mill 'polite smiles' can be a real indicator that the person you are talking to likes you and wants to continue the interaction. It is also worth noting that the orbicularis oculi does not work like a light switch, with a single set of on and off settings. You can have smiles of varying degrees of Duchenne-ness.
Dog owners who have heard the factoid that 'dog's cannot smile' may interested to note that dogs do have orbicularis oculi muscles, and will tense them when content, often while relaxing their jaw. This is not the only expression that a dog has that is equated with a smile, but it is one way that dogs smile, even though they lack the zygomatic major muscle to raise their lips into a traditional human smile.