For example it is stated that if supersymmetry breaking is soft then stability of gauge hierarchy can be still maintained.
In general, the terminology "soft" in particle physics refers to "low energy" or "low frequency" while "hard" refers to "high energy" or "high frequency". It's because hard radiation can penetrate and/or modify matter. Even in medicine, people are familiar with "soft X-rays" between 0.12 and 12 keV and "hard X-rays" between 12 and 120 keV - the latter can penetrate matter.
The term "soft SUSY breaking" has to be interpreted as one term. Of course, it is related to the softness explained above. The technical definition is that "soft SUSY breaking" is a term added to the Lagrangian that breaks supersymmetry, but only softly. This adverb means that the modification of physics at high energies is so small that we don't create any new divergent contributions to the mass of the scalars such as the Higgs.
The "softness" of this modification means that only the processes with low energies are being changed. High-energy scattering is not changed. It's analogous to the change that "soft radiation" makes on your body. It only affects the thin (and "soft") upper layers of the skin but doesn't penetrate to the bulk of your body, "hard" bones etc.
The same explanation for "soft symmetry breaking" holds more generally, not just for supersymmetry. When a symmetry is softly broken, the "hard" (high-energy) processes keep on respecting the symmetry while the low-energy, "soft" processes violate it.
One of the advantages of supersymmetry in phenomenology is that the bosons and fermions cancel their quadratically divergent loop corrections to the Higgs mass; soft SUSY breaking are non-supersymmetric terms that still preserve this cancellation. However, soft SUSY breaking obviously allows - and does cause - finite loop corrections to the Higgs mass. However, the quadratically divergent corrections are still absent.
If one looks which terms in the Lagrangian have this property, he finds out that it's generally the terms whose coefficients have a positive power of the mass - especially various mass terms for the superpartners of the known fermions and bosons. However, the terms that are generated from dimensionless (e.g. Yukawa or gauge) interactions with the Higgs or the gauge field can't be included among soft SUSY breaking terms.
In the minimal supersymmetric standard model (MSSM), there are about 105 new soft SUSY-breaking parameters. All of them influence the low-energy physics but preserve the cancellation of the divergences for the Higgs mass. In principle, the values of all these coefficients may be fully calculated from an underlying UV theory - such as an explicit compactification of string theory; in the full picture, the breaking of SUSY is always spontaneous. However, in the absence of knowledge about such a UV theory or vacuum, people partially give up and parameterize the low-energy manifestations of the supersymmetry breaking by the soft SUSY-breaking terms which are breaking SUSY "explicitly" and may be viewed as an effective description of the spontaneous SUSY breaking mechanism whose precise character in Nature - if any - is not understood at this moment.
I saw the words "soft breaking" used without relation to SUSY. I doubt that there is a strict definition of this, but naively it seems to mean the following thing:
A symmetry of your model can be broken either
- Spontaneously -- with some of your fields receiving non-invariant vacuum expectation value.
- Or explicitly -- by some terms in the Lagrangian that is not invariant under your symmetry transformation.
Explicit symmetry breaking is called "soft" if the terms introduced are of the dimension less than four.