We characterize by pattern avoidance the Schubert varieties for GL n that are local complete intersections (lci). For those Schubert varieties that are lci, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization requires only ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. Applications include formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties.