We introduce two new polynomials for hypergraphs, namely the edge cover polynomial and the edge decomposition polynomial, as the generating functions of the number of edge coverings and edge decompositions of a hypergraph, respectively. We study the algebraic and combinatorial properties of these polynomials and develop a recursive procedure for computing these polynomials based on the operations of vertex and edge deletion. It turns out that these polynomials can be used to enumerate the tilings (resp. coverings) of any bounded region of a lattice by 􏰀nitely many lattice animals. Examples include counting the tilings of a rectangle by polyominoes.