Strictness is typically an attribute of an authority, and implies that the person is not lenient towards subordinates who break a rule, but will punish wrongdoers without hesitation, up to to the full extent of their powers. A person can also be strict to him/herself, for example in fasting and other actions of self-discipline or penance.
Strict, as an adjective, can also apply to things. For example, a "strict rule" means that this rule is very important and must be strictly followed. Or a "strict order" is an order that must be strictly obeyed.
In BDSM, bottoms often like their tops to show and express strictness, as that symbolizes and shows strength, dominance and the readiness to punish harshly. Tops, in return, often enjoy being (or acting) strict towards their bottoms. Strictness can mean, for example, expecting immediate obedience to any command given, or (e.g. in 24/7 relationships such as domestic discipline), expecting that the rules given are followed, and taking appropriate punitive action if the sub fails to obey or follow the rules.
- rigor (US), rigour (UK)