Well the specific term "mammal," and its grouping of so many creatures under a common heading (by their teeth and the fact that they suckled milk from a breast — hence "mammal," which is derived from the Latin mamma, "breast") was done by Carl von Linné, best known as Linneaus, in his Systemae Naturae (10th edition, 1758). He saw them as a common "type" of creature, not a common lineage (Linneaus believed that taxonomy was fixed and rigid; he was a Creationist).

Prior to that, Linnaeus categorized mammals and many other creatures as simply quadrupeds (following Aristotle). Previous taxonomists classified animals under many different schemes, such as whether they had red blood (Aristotle). So we can pretty definitively give "mammals" as a specific classification (based on teeth and teets) to Linnaeus.

It is worth noting that the choice of mammae as the definitive quality of the grouping is a curious one, given that there are other choices that could have been made for the grouping (like the hollowness of their ears, or the presence of hair). The historian of science Londa Schiebinger has argued in a well-cited article that Linnaeus' choice of mammae is an odd one (considering that they are only functional in 50% of all species, and it is the only such animal class that Linnaeus used that was based on reproductive organs), and one that reflects a 18th-century European obsession with the idea of the magical powers of "mother's milk," and Linnaeus' own efforts as a physician to end the practice of wet nursing.

For more information on the development of the idea, as well as its antecedents, see Londa Schiebinger, "Why Mammals are Called Mammals: Gender Politics in Eighteenth-Century Natural History," American Historical Review 98, No. 2 (April 1993), 382-411.