Millikan's famous oil drop experiment didn't determine the mass of the hydrogen atom, but it did provide very important information for determining its mass: the elementary charge.
In the oil drop experiment, Millikan sprayed charged droplets of oil into a chamber of constant electric field. Some droplets fell, some rose, and some stood still. This is because the electric force in some cases balances out or overcomes the gravitational force on the droplets. From what he knew about the strength of the applied electric field, the density of the oil, and the strength of gravity, Millikan determined how much charge was on the droplets of oil that were standing still. He found that all the charges were integer multiples of a single number, which he dubbed the value of the elementary charge on an electron, -1.6 x 10-19
Now that we know the charge of a single electron (and thus a proton, because protons' charges are just equal and opposite in sign), we can determine the mass through a technique called mass spectrometry. This involves ionizing (giving charge to) particles and shooting them through a magnetic field. Magnetic fields act on charged particles by bending their path, turning them around in arcs. The arc is wider the heavier the particle is and the faster it's moving, but the arc is tighter if the magnetic field is strong or there is a lot of charge on the particle. Ultimately, we can determine the mass of a particle if we know how much charge is on it and by how much it curves in a given magnetic field. In a mass spectrometer that allows us to see the differences between more massive and less massive isotopes of carbon dioxide (the lighter particles curve more and the heavier ones curve less). That's shown in the attached illustration, borrowed from http://serc.carleton.edu/research_education/geochemsheets/techniques/gassourcemassspec.html
I hope this answers your question!
(published on 02/18/2011)