This is the first contribution of a sequence of papers introducing the notions of s-weak order and s-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers s. In this first paper, we concentrate purely on the combinatorics and lattice structure of the s-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the s-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the s-weak order to certain trees gives rise to the s-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the s-Tamari lattice can be obtained as a quotient lattice of the s-weak order when s has no zeros, and show that the s-Tamari lattices (for arbitrary s) are isomorphic to the ν-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the s-weak order will be studied in a sequel of this paper, where we introduce the notion of s-permutahedra.