Monotone triangles are a rich extension of permutations that biject with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains biject with monotone triangles; among these shellings are a family of EL-shellings. The weak order turns out to encode an action of the 0-Hecke monoid of type A on the monotone triangles, generalizing the usual bubble-sorting action on permutations. It also leads to a notion of descent set for monotone triangles, having another natural property: the surjective algebra map from the Malvenuto- Reutenauer Hopf algebra of permutations into quasisymmetric functions extends in a natural way to an algebra map out of the recently-defined Cheballah-Giraudo-Maurice algebra of alternating sign matrices.