In the old days, portraits were expensive, formal affairs. Cameras were big and bulky, with long exposure times. In the in 1800s, photographers had poles behind their subjects to help them hold their pose long enough to get a sharp photo. Their point of reference was the portrait painter, so they posed their photography subjects in stiff poses. You can still see these in the museums or collections online.
As time and technology marched on, smaller cameras allowed photographers to move outside of the studio and photograph their subjects in a more natural environment. Rangefinder and 35mm cameras really accelerated this movement. You can still find fixed studios, with their seamless background paper and perfect lighting set ups. Lots of families still use them, and many photographers like the ease of studios for their portraits. They can even be the only practical solution: how else can a school photographer complete her task of depicting hundreds of young kids in a setting their parents will appreciate and still keep costs reasonable? Since I don't have a similar task, I prefer to get away from this and put my subjects in a “real-world” setting.
This type of people photograph is fun for you and your subject because you catch them doing things they enjoy or in their familiar surroundings. They can be called “location photography,” “environmental portraits,” or many other titles photographers like to invent to make our work seem more unique. Whatever you call it, this photography can seem more personal, providing insight to your subject; it can help tell a story; and lead to your photograph being the one they like to display.
Where you go is up to you - that's what is so rewarding about this kind of people photography. Make the photographs of them participating in their hobbies, games, or professions. Develop your own locations with several different backgrounds to use when making outdoor portraits. Go out and capture little slices of their world and what's important to them.
You can still make portraits using this technique. When I think of the word “portrait,” I think of a head and shoulder shot, where the person is the center of interest and takes up the majority of the frame. You can certainly do that in any location.
While most of the photos I've shown are casual, you can still incorporate this type of photography in posed situations. You can photograph your subjects in the existing light or bring along flashes and reflectors if you want more control of the light. I carry some of the multi-use reflectors in my truck, so they're always ready. (Don't listen to my wife who thinks I'm just driving a cluttered vehicle.)