Underpinning allows you to retain all of your basement square footage since there is no need to build a bench wall inside your basement. It's a tricky and time consuming job, and one not left to fly by night contractors.
Underpinning involves excavating and pouring new footings to the depth where your new basement floor will be. This is done in sections, and the load of the foundation is carefully transferred and supported as each section is completed. It's a slow and cautious process as you're basically undermining the integrity of the foundation during the basement lowering process.
There are numerous advantages to this model. First and foremost the basement square footage remains the same but you get the extra headroom to make it new livable area in your house, thus increasing value of your biggest investment significantly. Moreover, if you had the slightest doubt about the state of the original foundations that concern's gone forever, it will be replace with a brand new one which will last for generations.