The halo Turner is referring to is the halo of the Milky Way galaxy, the region beyond the central disk of stars and dust. If current theoretical models are correct, there's a massively massive pool of dark matter - perhaps as big as 1 million light-years across - that envelopes the visible galaxy, which is about 100,000 light-years in diameter.
Behind the AMS numbers is an 80-year-old mystery about why our galaxy - and the universe for that matter - hangs together because despite the apparent plethora of stars, galaxies and gas, there is simply far too little of it to gravitationally bind it together.
BIG PIC: Hubble Builds 3-D Dark Matter Map
Physicists estimate that visible (i.e. detectable) matter accounts for a mere 4 percent of the universe's contents. Dark matter, which is not dark as in "black" but dark as in undetectable with electromagnetic radiation, comprises about another 24 percent. The rest is an even more exotic and less-known force called dark energy.
One idea about dark matter is that even though we can't, by definition, detect it directly, we can scout for its footprints.