In mathematics, local generally means that a property or behaviour at a point depends only on that point, or one points close to it. Often, it means locally determined -- some global property is determined completely by the way things behave locally. In addition to local being used as an adjective, locally is also used as an adverb, typically as a modifier to existing properties to indicate that these properties are only satisfied locally.
In ring theory
In both commutative and noncommutative algebra:
In group theory
Local subgroup: A subgroup of a group that occurs as the normalizer of a nontrivial solvable subgroup.
Primary subject wiki entry: Groupprops:Local subgroup