I don't really think that there is really a name for this effect, at least officially.
A long time ago when I still visited this site, I found a tutorial for creating this effect in Photoshop. They called the effect the "Pleasantville Effect." I have never heard of this effect anywhere else. The tutorial link is here: http://www.albinoblacksheep.com/tutorial/pleasantville
But, if you read the tutorial, you'll find that this effect is very simple. I'll summarize it in general terms that you might apply to when you use a free program like Paint.NET (download link here: http://www.download.com/3000-20-10338146.html?part=undefined&subj=dl&tag=button ). (Side note: if you download Paint.NET you will also have to download something called the .NET (dot net) framework from the Microsoft website at Microsoft.com. Just search it under downloads.)
Basically, you start out with a color photo. Use a selection tool to carefully draw a selection shape around the thing you want to be color while everything else is black and white.
Then, copy the selection, make a new layer in the picture, and paste it in the new layer while making sure it's in the same spot as the original selection. Then, go to the old layer with everything else but the thing you want to be color, and run it through a filter that makes everything you run through it black and white.
I suggest you read the tutorial because the way you make the so-called "Pleasantville Effect" in Photoshop is very similar to the way you make it in any advanced image editor like Photoshop or Paint.NET or Paint Shop Pro.