One of the methods that I have liked doing lately is what is called high key studio photography. You may have seen this before it is when you see those images where the subject seems to just have perfect clear white behind them. This is a great method especially for kids. It provides colorful vivid images that really show the personality of the children. For a basic setup there are a minimum of 5 key elements in your studio kit.
1. A background stand with white background. “I uses a fabric store bough muslin however you can also use seamless paper or any other solid background material like bleached canvas
2. A boom and a stand. Alternatively you can have a light stand over to one side with the flash directed at the background. However this will not evenly distribute the light across the back drop.
3. A Key light Main with a light modifier. Umbrella, soft box so on.
4. Speed light flashes that are able to shoot in manual the higher the guide number the better, at least over 100 feet @ 100 ISO.
5. A trigger system radio, optical, or sync cables
With kids since they are most likely to move and are going to be near the floor the flash must be above them so that it is out of the way. You also want to put your key main light either right next to you or behind you to protect it from getting knocked over. If you are just doing head shots on a model or a older subject you can put the rear backdrop flash on the ground directly behind the subject pointing at the background also.
The goal is to have the background as bright as possible. Normal I try to have the background at full power 1/1 if possible. The problem with this is recycle times. This will give a nice edge light on the subjects hair and shoulders. The power on the main strobe varies on your ISO, aperture, or shutter speed on your camera or the light modifier you are using. If you bouncing the flash it needs to be a little higher light output than if you are using a shoot through umbrella. Test your setup before hand and adjust if needed.

Strobist:
-Vivitar DF 400 MZ on boom on backdrop.
-Nikon SB-24 r shot into 30 inch white shoot through umbrella with 1/4 cto warming gel.

Strobist:
Vivitar DF400MZ 1/1 power on boom pointing at backdrop
Nikon SB24 1/1 power 1/4 cto gel shoot through umbrella upper camera right
Lighting diagram: farm3.static.flickr.com/2726/4104706380_dabe101457_o.png

Strobist:
-Vivitar DF 400 MZ 1/4 on boom behind subject on backdrop.
-Nikon SB-24 1/2 power shot into 40 inch sliver umbrella directly above and to the left of the camera with 1/4 cto warming gel.
She took my umbrella stand and was using it as a tripod and a microphone.

Strobist:
-Vivitar DF 400 MZ 1/8 on boom behind subject on backdrop.
-Nikon SB-24 1/2 power and Vivitar 2500 1/1 power rubber banded together shot into 40 inch sliver umbrella directly above and to the left of the camera with 1/4 cto warming gel.

I dont have many pictures of me. I just got done with a shoot for lillie and I had my speed lights still out so I decided why not. I got my trusty Canon A-1 film camera and focused on the chair with my 20d and hit the self timer.
Strobist:
Vivitar DF400MZ 1/1 power on boom pointing at backdrop
Nikon SB24 1/1 power 1/4 cto gel shoot through umbrella upper camera left