template structures: known [protein|rna|other biomolecule] structures that will be used in homology-modeling and threading/fold recognition to generate a structure for your input sequence based on sequence similarity

non-redundant template structures: a set of all known structures that are deemed to be unique in the set of all known template structures. Uniqueness is determined according to some similarity measure (e.g. backbone RMSD of superposed 3D structures, sequence similarity of primary structures, topology differences of secondary structures etc). Non-redundancy allows you to limit your search space, otherwise you'd try using the same (or very similar) template structure all over again, which increases overall runtime and complexity.