Simply put biochar is charcoal made by burning biomass and using it to enhance soil and/or capture CO2. The idea is that the charcoal improves the soil's carbon content and functions as a carbon sink for atmospheric CO2. This would be especially useful for depleted soils. The idea of using charcoil to enhance to soil was derived from the Terra Preta found in the Amazone. Terra Preta is very dark and rich soil with high content of charcoil that supposedly was created by Amazone natives between 450 BC and 950 AD. However, research has to point out how much of the idea behind biochar is actually true.
You can either buy Biochar or you can make it yourself via pyrolysis, which is burning organic materials with very low levels of oxygen. Here is one movie on how to make it, but you can find many other movies or webpages on the net.
Biochar is currently used in all sorts or places - farm fields, gardens, indoor plantpots - but since we don't know much about its effect yet, its usage is mostly experimental. Raw biochar needs to be mixed with compost and water before adding it to your garden or else it will draw nutrients and water away from your plants. How much to add depends on the type of soil you have and possibly on what you want to grow, so you may want to do a soil analysis first.
There is quite a bit of professional research going on, as well as people experimenting themselves (just search the internet). Personally I found this link very informative and useful.
Note that there is also a Gardening SE question on using charcoal as soil amendment with some interesting answers.