A mathematical property holds locally if is true near every point. In many different areas of mathematics, this notion is very useful. For instance, the sphere, and more generally a manifold, is locally Euclidean. For every point on the sphere, there is a neighborhood which is the same as a piece of Euclidean space.
The description of local as "near every point" has a different interpretation in algebra. For instance, given a ring and a prime ideal , there is the local ring , which often is simpler to study. It is possible to understand the original ring better by patching together the information from the local rings.
What ties all the notions of local together is the concept of a topology, a collection of open sets. For a submanifold of Euclidean space, or for the set of ideals of a ring, the topology is chosen as is appropriate.