Nearly the entire mass of an atom is comprised by its nucleus. The hydrogen atom is no exception. The mass of a typical hydrogen atom (known as protium, P) is 1amu, which tells you that it consists of 1 proton. Of course, there's also 1 electron, but this electron's mass doesn't come into play (an electron's mass is about 0.0005amu).
On the Periodic Table, the atomic mass of hydrogen comes in at about 1.00794. This is a result of the small of percentage of all hydrogen out there that consists of more massive isotopes. H-2 (deuterium, D) has 1 neutron; H-3 (tritium, T) has 2 neutrons. The 1.00794 is a weighted average of these relative abundances.
I think the answer is simpler than that. The question may simply be about the fact that the measured mass includes an electron, but the nuclear mass does not. It depends a bit on how you interpret this question.
Anyway, the proton has about 1836 times the mass of an electron.
So, the mass of a hydrogen atom is almost the same as the mass of the nucleus, as mentioned above.
Incidentally, the definition of atomic mass (which is slightly different from "nuclear mass") is not quite as straightforward as some people may think.
The atomic mass of "protium" is, in fact, not 1 amu, but about 1.00782 amu.
So, for the atomic mass of hydrogen, the mass of the electron is actually slightly more important than the effect of the small amounts of heavy isotopes.
You may want to click on the link below for more details on the atomic mass of hydrogen.
(Strictly speaking I think you should call that the "relative atomic mass" when talking about all the isotopes together in a sample, but that's a bit pedantic here.)
The Wikipedia page "Isotopes of hydrogen" is useful too.
Also, there's a link below to a "related question".