Shooting portraits of people is a difficult thing, as you have to manage not only technical parameters involving the camera and lighting, but also the person's posing and expression. This excellent video essay reminds us of what's probably the most important aspect of any portrait.
Coming to you from Light Club, this great video essay looks at the work of some master portrait photographers and what they all have in common, namely the priority they place on captivating the viewer with the subject's eyes. These are naturally one of the most important features of almost any good portrait, as they draw the viewer in and help them empathize with the subject, as this mimics human interaction and thus, makes the viewer (at least subconsciously) transcend the medium of the photo and feel a genuine connection with your subject. It's well worth taking a couple of minutes to view the examples shown in the video; notice how regardless of the different photographic styles, the one constant is the careful attention each photographer paid to the eyes.
And if you really want to dive deep into the art of taking eye-catching photos, check out "Peter Hurley: Perfecting the Headshot."