This refers to the number of "de Sitter" vacua, in lay terms the number of "different versions of physics", and the estimate has been growing. As of 2003 the estimate was at least 10250. Then for the past few years we have been mostly hearing 10500 and lately I hear it's probably more like 101000.
The prime source on this is the top-cited paper here:
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+DK+STRING+MODEL+AND+DATE+%3E+2002+AND+TOPCITE+500%2B&FORMAT=www&SEQUENCE=citecount%28d%29
This was the paper in 2003 that originally brought out the 10250 number. Judging by how often it has been cited as a reference in other string research it is the most important string paper that has come out in the last eight years. Between when it came out in 2003 and the present it has been cited 1336 times.
The simple intuitive reason there are so many different versions of empty space (each giving a different version of physics) is that in string theories there are a lot of extra dimensions that have to be rolled up.
And that means there are a lot of different possible ways to roll them up, which first? in which order?, and which inside which? There are also different ways to wrap them so that they will be stable.
These different ways of doing up the extra dimensions lead to string theories with differerent versions of empty space and therefore different versions of physics (forces, particles), with no logical or mathematical principle that makes anyone particular version stand out.
A number of the top people have been shifting their research interest to topics that do not require a lot of extra dimensions, and don't face this problem of the multiplicity of vacuums. It's an interesting trend to watch.