the hierarchy problem
Two energy scales set by nature sit absurdly far apart. The scale of the weak force, where the Higgs and the W and Z bosons live, is around 100 GeV; the Planck scale, where gravity becomes strong, is around 10^19 GeV. That is a gap of seventeen orders of magnitude, and there is no obvious reason for it. Why is gravity so incredibly feeble compared with the other forces, or equivalently, why is the Higgs boson so light? That question is the hierarchy problem.
The problem is sharpest for the Higgs mass. In quantum field theory, the mass of a particle receives quantum corrections from every particle it couples to, and for a scalar like the Higgs these corrections are quadratically sensitive to the highest energy scale in the theory, growing like the cutoff squared. If that cutoff is the Planck scale, the natural size of the Higgs mass-squared would be about 10^34 times too large. To get the observed value of 125 GeV, the enormous bare mass and the enormous quantum corrections must cancel to about one part in 10^32, a fine-tuning that looks like an unexplained conspiracy rather than an accident.
The hierarchy problem is not an inconsistency, everything computes fine, but a naturalness problem: a dimensionless number is inexplicably tiny with no symmetry protecting it. Proposed cures introduce new physics near the weak scale to cancel the dangerous corrections: supersymmetry pairs each particle with a partner whose contributions offset it, while extra-dimension models lower the true Planck scale. The Large Hadron Collider was expected to find such new physics, and its absence so far has deepened the puzzle rather than solved it, keeping the hierarchy problem one of the central open questions of particle physics.
It is like balancing a pencil on its tip and finding it still upright after a billion years: not impossible, but so unlikely without a mechanism that you suspect something is holding it. In the same way, the Higgs mass staying at 125 GeV despite Planck-scale quantum corrections looks like a fine-tuning that cries out for a hidden cause.
The Higgs mass survives huge quantum corrections only by a 1-in-10^32 cancellation.
The hierarchy problem is a problem of naturalness, not of mathematical consistency; the Standard Model works, it just requires an unexplained fine-tuning. Supersymmetry was the favored cure, but its non-appearance at the LHC so far has sharpened, not settled, the puzzle.