Hardness and softness are terms used to describe which aspect of acidic behaviour?
The terms softness and hardness are used in HSAB theory to distinguish Lewis acids and bases by charge density, polarizability, electronegativity, and, in the case of molecular compounds, HOMO and LUMO energies.

Charge density is inversely proportional to atomic/ionic radius and proportional to oxidation state (in case of a neutral atom) or charge (in case of an ion). Hence charge density increases with decreasing radius and increasing oxidation state.

Polarizability is the tendency of the electron shell of an atom/molecule to become deformed in an electric field, which can originate e.g. from the charge of an adjacent ion or the partial charge of a dipole. An electron cloud of an atom or ion is more easily deformed with increasing volume, as it is more diffuse, and its outer electrons tend to be more loosely bound. Therefore, larger atoms and molecules tend to be more polarizable, and higher polarizability is often accompanied by lower charge density or oxidation state.

In the case of molecules, when a Lewis acid reacts with a Lewis base, electrons from the highest occupied molecular orbital (HOMO) of the Lewis base are transferred to the lowest unoccupied molecular orbital (LUMO) of the Lewis acid. How well this transfer of electron density works depends on the relative energies of the acid LUMO and the base HOMO. The stability of the reaction product will increase with the energy difference between its HOMO and LUMO (source).

The HSAB theory predicts that hard Lewis bases react preferentially with hard acids, and that likewise soft bases form stronger bonds with soft acids. For example, $\ce{Li+}$ and $\ce{BF3}$ are hard acids and $\ce{F-}$ and $\ce{NH3}$ are hard bases due to their small size and high charge density/low polarizability of the ions. $\ce{BF3}$ therefore readily forms Lewis acid-base adducts with fluoride and ammonia, and $\ce{LiF}$ is less soluble in water than $\ce{LiCl}$ and $\ce{LiBr}$. $\ce{Ag+}$ is an example for a soft acid (large ion with low charge), and the halide ions become increasingly softer bases with increasing ionic radius. Silver therefore forms stronger bonds with heavier halides like $\ce{I-}$ and $\ce{Br-}$, and this also explains the decreasing solubility of the silver halides in water, which is classified as a hard solvent. In the case of soft acids and bases, the difference of electronegativity tends to be smaller, and the bonds have more covalent character (e.g. silver iodide) than those between hard acids and bases (e.g. lithium fluoride) with larger difference in electronegativity (source).

The shift of reaction equilibria can also be predicted with this concept, like in the following example (reference):

$$\ce{HgO + H2S \rightleftharpoons HgS + H2O}$$

This equilibrium will be shifted to the right side, because $\ce{Hg^2+}$ as a soft acid prefers to bond to the soft base $\ce{S^2-}$. Likewise, oxygen as a hard base forms stronger bonds with hydrogen (hard acid).