According to my old college dictionary, the one still within arms reach of my desk, a portrait is defined as, "1. a drawn, painted or carved picture of something. 2. a representation of a person, esp. of his face. 3. a description, dramatic portrayal, etc. of a person".
A portrait can be many things. You may think of the classic studio portraits or the shots we all had done as kids in school; maybe you like location portraits that show a person in their surroundings or portraits of your dog or cat. Portraits can be of things, too, like flowers, icebergs or even your home town.
The best portraits tell you something about a person, place or thing. It reveals a snippet of what they are like, what their story might be. But portraits also reveal something about the person taking the photo--it's a collaboration between photographer and subject.
I love portraits of all kinds, B&W studio shots, environmental portraits, tight face shots, action shots, moody "feeling" shots; portraits of kids, old people, CEOs or someone on the street. The possibilities are endless.
Ultimately, it comes down to being interested in people and places, loving the medium of photography and the desire to share a story.