limit
A limit is a single object that optimally summarizes a whole diagram of objects and arrows, the way a product summarizes a pair. Given a configuration of objects with prescribed relations between them, the limit is the universal object that maps compatibly to all of them, respecting every arrow in the diagram. It is the categorical home for products, intersections, kernels, pullbacks, and inverse limits, all of which are limits over particular shapes.
Precisely, fix a small category J (the shape) and a functor D : J -> C (the diagram). A cone over D from an object X is a family of morphisms X -> D(j), one for each object j of J, that commute with all the arrows D sends. A limit of D is a universal cone: an object lim D with a cone such that every other cone factors through it by a unique morphism. The universal property again forces uniqueness up to unique isomorphism.
Limits are limit-of-shape sensitive: a product is the limit over a discrete diagram, an equalizer the limit over two parallel arrows, a pullback the limit over a corner, and an inverse limit the limit over a descending chain. A category in which all small limits exist is called complete; Set, Grp, Ring and Top are all complete. A key fact is that right adjoint functors preserve all limits, a frequently used computational shortcut.
The ring of p-adic integers Z_p is the inverse limit of the system ... -> Z/p^3 Z -> Z/p^2 Z -> Z/pZ, a limit over the descending chain of quotient maps. An element of Z_p is a compatible sequence of residues, exactly what the limiting cone records.
An inverse limit is a limit over a descending chain.