So once we have measured the current expansion rate of the universe -- i.e., the Hubble constant -- the next step is to figure out how much mass is in the universe. Why is this important? Mass alters the expansion of the Universe through its gravitational effect on space. If there is enough mass in the universe, its gravity can actually act to halt the expansion, and perhaps even cause the universe to recollapse on itself. How do we characterize the amount of matter in the universe? For a moment, let's ignore the possibility that the universe's expansion may be accelerating. We can then define a "critical density" -- the density of mass which is just sufficient to halt the expansion. We then parameterize the mass density of the universe by a term called Omega_M, which is the density of the universe divided by the critical density:
If OmegaM = 1, the universe has just enough mass to halt the expansion. If OmegaM < 1, the universe has less mass and would keep expanding. If OmegaM > 1, the universe gas more mass and will actually end up recollapsing on itself.
So what is OmegaM? This is a very hard parameter to measure, particularly since so much of the mass of the universe seems to be in the form of "dark matter" -- material which does not emit light. There are various ways of measuring the mass of the universe, by looking at the dynamics of galaxies and galaxy clusters, or even by measuring the way mass bends the light of background galaxies. With considerable uncertainties, the current "best estimate" of the mass density of the universe is in the range 0.2 < OmegaM < 0.5.
One other constraint is that most current theoretical models of the universe predict that the universe should have Omega = 1. Of course, this may not be the way the universe is...