Donnellan's main claim in this paper is that there is something wrong with both Russell's and Strawson's account of descriptions, because both of them account for at most one of the uses of descriptions that Donnellan distinguishes. That said, in many ways one might think of Donnellan as trying to build on Strawson's central point about the importance of reference as an act.
The central contribution of this paper is the distinction between attributive and referential uses of descriptions. One uses a description attributively when one wishes to speak about whom- or whatever satisfies it. By contrast, one uses a description referentially when there is a specific object about which one wishes to speak, and one only uses the description in order to help one's audience identify it. Only in the latter case would it make sense to ask, "Who do you mean?" or "Who are you talking about?"
Make absolutely sure you understand this distinction. One good way to do that is to come up with some examples of your own to illustrate it. Better yet, pay attention as you read non-philosophical material, and see if you can find some good, independent examples of each use. If you're having trouble, email me.
And, as Donnellan is at pains to emphasize (pp. 285ff), this is a feature of uses of descriptions, not a matter (as Strawson seems to imply) of where they occur in a sentence (as subject or as predicate). Thus, Donnellan suggests that someone who uttered "Smith's murderer is insane" could be using the description either way, depending upon what sort of thing they were trying to say.
Donnellan uses a variety of locutions in trying to capture what is special about the referential use: He talks of the thing that is "meant"; of something that the speaker has "in mind". Are these helpful? Why or why not? Or better: In what ways are they helpful and in what ways not?
Perhaps the most important difference between these uses, however, appears when we assume that the description is "improper", i.e., that there is no unique object that satisfies it (pp. 286ff). Donnellan claims that, if we utter "The F is G", using the description attributively, and it is improper, then there is no sense in which anything has been said to be G. But if the description is used referentially, then one might still have managed to refer to something and to say of it that it is G. This is a phenomenon that neither Strawson nor Russell seems to have envisaged.
Donnellan notes (pp. 288-9) that, although there is in both of these cases some sort of implication or presupposition that something fits the description, what is implied or presupposed depends upon the kind of use being made. In the attributive case, the implication or presupposition is general: One is implying or presupposing that something fits the description uniquely. In the referential case, by contrast, one is implying or presupposing that some particular thing—namely, the thing about which one wishes to speak—fits the description uniquely.
In section IV, Donnellan extends these claims in two directions. First, he argues that the question whether a description is used referentially or attributively cannot be reduced to the question whether the speaker knows of some particular thing that it fits the description uniquely (p. 289). Second, he argues that it is possible to use a description referentially even if one believes that the object to which one wants to refer does not fit the description, or even that nothing does (pp. 289-90). As Donnellan notes in section V, this latter point qualifies the earlier one about presuppositions (p. 291): In the referential case, it would seem, there is only a "presumption" that the object about which one wishes to speak fits the description.
In section VI, Donnellan argues that his results are inconsistent both with Russell's theory and with Strawson's. Can you briefly summarize why? Which of the two theories would you suppose would have a harder time accomodating the referential–attributive distinction?
In the last couple sections of the paper, Donnellan attempts to say something about the nature of this distinction he has isolated. In section VII, he denies that it is any sort of ambiguity, either syntatic (e.g., a scope difference) or semantic (a difference in the meanings of the words, as with "bank"). He suggests we might think of the ambiguity as "pragmatic", a matter of the speaker's intentions, but it is not clear what this might mean.
The issue here is a very general one. Both Russell and Frege, one might say, tie the proposition the speaker expresses very tightly to the meaning of the words the speaker uses. Strawson to some extent tries to open up some space between these, for example, in the case of incomplete descriptions, such as "The table is covered with books". One might suggest that Donnellan is going even further in this direction. How so?
In section VIII, Donnellan suggests that we might understand the referential use in terms of a notion of "say[ing] something true about someone" (p. 298). Thus, even if Jones did not murder Smith, we might say, in the original example, that we referred to Jones and said of him that he was insane. And it seems as if Donnellan thinks this is the best we can do. He seems to want to deny that any "statement" is made in such cases (i.e., that any definite proposition was expressed), but seems to want to insist that the best we can do is to use the said-of locution. This sort of idea is developed yet further in the final section, in which Donnellan suggests that there is something fundamentally right about Russell's idea that, when we genuinely refer to something, we do so without thereby ascribing any specific properties to it. (Of course, having referred to it, we presumably will then ascribe some properties to it.)
Suppose that the best we can do, in cases of referential use, is to say that the speaker said of a particular object that it is G. Suppose, moreover, that in characterizing what the speaker said, we ourselves can refer to the object to which the speaker was referring using any tool at our disposal: any name of the object, e.g. It would be natural to understand this as a challenge to Frege. Why? (Hint: Consider a case in which, say, "the morning star" is used referentially.)