1. In abstract algebra, irreducible is an abbreviation of irreducible element; for example an irreducible polynomial.
2. In representation theory, an irreducible representation is one with no nontrivial subrepresentations.
3. In the theory of manifolds, an n-manifold is irreducible if any embedded (n-1) sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.
The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connect sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over S^1 and the twisted 2-sphere bundle over S^1.
A theorem of 3-manifold theory is: every compact, connected 3-manifold has a prime decomposition, i.e. can be written as a connected sum with each summand being prime. This prime decomposition is also unique (up to homeomorphism of summand). [Again, we must be working in either the differentiable or piecewise-linear category]
4. In algebraic geometry, an irreducible algebraic variety W is one that cannot be written as a union of subvarieties U and V, except when one of those is contained in the other.