Good reduction of a smooth proper algebraic variety over a discretely valued field means that the variety extends to a smooth proper model over the valuation ring. The special fiber of that model is then a smooth proper variety over the residue field. Good reduction of an abelian variety is expressed by an abelian scheme over the valuation ring. When descent and a finite local extension are allowed, they must be specified as part of the assertion.
For example, an elliptic curve with an integral Weierstrass equation and unit elliptic discriminant has good reduction. Its reduced equation remains nonsingular. For abelian varieties, good reduction allows specialization of Hodge cycles to the cohomological realizations used in Milne's rationality conjecture.
