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In theoretical physics, a hierarchy problem is a confusing observation that two fundamental quantities with the same units have vastly different values, and therefore the naïve calculation based on dimensional analysis can lead to incorrect results. In other words, it is an apparent paradox: while all the numbers originating from a fundamental theory should naïvely be comparable to 1, we observe numbers in Nature that are much smaller (or much greater).
More specifically, in particle physics, the hierarchy problem is the big question why the typical energy scale associated with the electroweak symmetry breaking - roughly, the typical size of all masses of elementary particles - is so much (<math>10^{15}<math> times) smaller than the Planck energy. More technically, the question is why the Higgs boson is so much lighter than the Planck mass, although one would expect that the large (quadratically divergent) quantum contributions to the Higgs boson mass squared would inevitably make the mass huge, comparable to the Planck mass.
This is just one of the many cancelations to the quadratic divergence to the mass squared of the Higgs boson which occurs in MSSM. The most popular theory - but not the only proposed theory - to solve the hierarchy problem (i.e. to answer the question) is supersymmetry. But this only explains how a tiny Higgs mass can be protected from quantum corrections, but not why it is small but still nonzero in the first place. Put another way, why are we so close to a second order phase transition?
Of course, since we don't know for sure what physics above the TeV scale or so is, or even lower, it might turn out there is some new physics which would make the whole problem of the hierarchy problem moot.
In the Standard Model, there is the problem as to why masses of the three fermion generations are different by several orders of magnitude.
There is also the problem as to why neutrino masses are so tiny (see neutrino oscillations) compared to an electron, but this can be solved by the seesaw mechanism in GUT theories, among others.
In cosmology, if we are to trust current observations in favor of an accelerating universe, there is the problem as to why the cosmological constant is tiny but nonzero and why quantum corrections do not give a much larger value for it. SUSY can only protect it up to the SUSY breaking scale, which is still way too large.
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