Cosmological arguments begin with the bare fact that there are contingently existing things and end with conclusions concerning the existence of a maker with the power to account for the existence of those contingent things. Teleological arguments (or arguments from design) by contrast begin with a much more specialized catalogue of properties and end with a conclusion concerning the existence of a designer with the intellectual properties (knowledge, purpose, understanding, foresight, wisdom, intention) necessary to design the things exhibiting the special properties in question. In broad outline, then, teleological arguments focus upon finding and identifying various traces of the operation of a mind in nature's temporal and physical structures, behaviors and paths. Order of some significant type is usually the starting point of design arguments. Various advocates have focused on different types, levels and instances of order, have suggested different logical connections between order, design and designer and have pursued different levels of rigor—from Bayesian formalisms to the deadly serious whimsy of G.K. Chesterton:
So one elephant having a trunk was odd; but all elephants having trunks looked like a plot. (Chesterton 1908, 106-7)
Design-type arguments are largely unproblematic when based upon things nature clearly could not or would not produce (e.g., most human artifacts),