There's others out there with much more knowledge than me.
In my mind, they're an efficient way of increasing bearing capacity, reducing settlement and eliminating the risk of liquefaction. They do transmit vibrations to nearby structures, so they're a little better for green field type work, than congested sites.
One bit of warning. Geopiers (and Rammed Aggregate Piers or RAP)is a trade name, religously protected by the parent company. If you name Geopiers on your drawings or specifications, that's what you'll get. Other soil improvement systems are very similar, but can't bid on a Geopier Project. So if your Geotechnical Engineer is recommneding Geopiers, have them use a more generic term like stone column densification, so you don't get boxed in.
Take a look at the March 2012 issue of Structure magazine for a "Soil Improvement Primer" http://www.structuremag.org/article.aspx?articleID=1411 which explains these things much better than I can.

Rammed Aggregate Piers - Geopiers - are best suited for soils with at least 15% fines (-200).  That is, at least some cohesion, and the drilled hole stays open without sloughing.
Generally, ML-CL will see the greatest improvement in bearing capacity and settlement control.
Very different than vibratory methods.  Tapered mandrel puts the soil in condition where the radial "prestress" is greater than the vertical.  Changes the stress state of the soil (Mohr-Coulomb failure line).
Talk directly with the local Geopier licensee.
For sandy soils, not advised, since casing is required, and the cost (and risk) doubles.
Jed is correct about the parent company and trade name use.
Stone columns are ideal for sandy soils, not so good for clays.