The heart of Putnam's paper is the claim that "meaning ain't in the head" and the example of Twin Earth that is designed to support that. Both are presented in the fourth of the sections included in Chalmers (pp. 584-587), and it is likely to be the focus of our discussion.
The preceding sections of the paper set up the issue. One of the key ideas is the distinction between intension and extension; you encountered this in Place and Smart (the example of "the Morning Star" and "the Evening Star," see Chalmers, p. 62, is a standard way of introducing it). Another key idea is "methodological solipsism."
The idea of "division of linguistic labor" is another important idea in Putnam's discussion. Others have taken these sorts of considerations further. (The most notable development of this kind of idea is the paper by Tyler Burge that follows Putnam's in the Chalmers anthology; in particular, the example in Burge's section IIa, pp. 599-601, points to conclusions analogous to Putnam's.)
In the remainder of the paper, Putnam tries to take account of the implications of these ideas for the concept of meaning. This part of the paper is less important for our purposes and you may find parts of it somewhat technical (especially the section on pp. 588-591--though the only symbol there is Putnam's use of ≡ for "if and only if"). Still you may find parts of this material useful in summing up Putnam's point; the next to last section, pp. 593f, may be the most useful in this way.
Part of your assignment for next week is to look through the Chalmers anthology to find selections you'd like to read after Thanksgiving. Each of Chalmers's section introductions (i.e., pp. 1ff, 197ff, 473ff, and 653) gives a brief summary of the contents of each of the papers in that section. Look for a couple of them that you'd like to discuss in the last couple weeks of the course.