Russell's Theory of Descriptions can be best understood as a response to the shortcomings in Frege's philosophy. He sought a theory of language which, firstly, would avoid using or relying on any Fregean concept of Sinn and, secondly, would solve the "problem of existence" in a way that allowed the truth-value to be determined for all sentences without committing us to the existence of any reference less constituents.
Central to both Frege and Russell's approach to language was the belief that "the apparent grammatical form of a sentence can mislead us about the hidden logical form of the proposition the sentence expresses." For Russell one of the key distinctions that Frege failed to make in attempting to logically analyze ordinary language was that between logically proper names and descriptions. Russell believes that many of the terms that Frege had considered to be names were not in fact logically such, but were rather sorts of "truncated descriptions."
Russell argues that the meaning of a logically proper name is "the individual thing it designates." "A logically proper name is meaningless unless there is some single object for which it stands." In the majority of cases, however, Russell believes that phrases of "the so-and-so" are in fact descriptions.
(With regard to descriptions he makes a further distinction, namely: "An indefinite description is a phrase of the form 'a so-and-so' and a definite description is a phrase of the form 'the so-and-so' (in the singular)." Russell's Theory of Descriptions deals with the former, and henceforth I shall refer to them as descriptions simpliciter.) A description is recognized by its form, "and not whether there is a definite individual so described." Russell defines it in opposition to a name, and gives the following differences: + A description is a complex symbol. A name is a simple symbol.
+ The meaning of a description is fixed and determinate when the meanings of its terms have already been fixed, i.e. "you would understand the meaning [of it] if you had never heard it before." In contrast, you would not understand the meaning of a name if you had never heard the word before "because to know the meaning of a name is to know what it is applied to." Russell's analysis allows that traditionally problematic sentences can be solved by rewriting them in a logically appropriate grammatical form. Let us reconsider the example of S: "The King of France is wise." According to Russell, anyone who asserted S would, in fact, be logically asserting the following three propositions: At least one person is the King of France.
At most one person is the King of France.
Whoever is the King of France is wise.
In other words, while D: "The King of France" is the grammatical subject of S it is not the logical subject. Frege overlooks this distinction. D is not a logically proper name that stands in a subject-predicate relationship with "wise". In fact, logically speaking, S is not a subject-predicate sentence at all, but "a complex kind of existential proposition, part of which might be described as a 'uniquely existential' proposition." Since S is a conjunction of three propositions it follows that if one of them is false, then S is false. Thus Russell demonstrates how the truth-value of S can be determined without necessarily positing any existing object denoted by D.
As I have portrayed it thus far, Russell seems to have been quite successful in presenting a solution to the basic problems of identity and reference outlined at the beginning of this essay, while avoiding the problems and shortcomings created by the basic Fregean theory of objective sense and truth-value-determining-reference.
I wish to conclude by briefly examining the well-known critique of Russell offered by Strawson. According to Strawson, Russell recognizes only two ways in which sentences which appear from their grammatical structure to be about some particular thing can be significant. The first is that their grammatical form should be misleading as to their logical form, and that they should be analyzable, like S, as a special kind of existential sentence. The second is that their grammatical subject should be a logically proper name, of which the meaning is the individual thing it designates.
In contrast Strawson denies that an expression used in "the uniquely referring sense" (e.g. a singular subject predicate sentence) falls into either of the two classes. He contends that there are no logically proper names and no descriptions in the Russellian sense. To support this claim Strawson introduces certain distinctions that he thinks Russell incorrectly overlooked.
Firstly, he distinguishes between an expression used in the uniquely referring sense and a sentence beginning with such an expression. Henceforth I shall term both respectively expression and sentence simpliciter. Secondly, he distinguishes: (A1) a sentence (A2) a use of a sentence (A3) an utterance of a sentence (A1) refers to the sentence itself, for example S: "The King of France is wise", which can be uttered on various occasions by various speakers. There are, however, obvious differences between the occasions of the use of this sentence S. If two men uttered S, one in the reign of Louis XV and one in the reign of Louis XIV, each made a different use out or the same sentence.
Thus (A2) refers to the particular use of a sentence.
Finally, (A3) refers to the different individual utterances of S which make the same use of it, for example two men in the reign of Louis XV simultaneously uttering S.
In these terms, Strawson identifies the root of Russell's mistake in his failure to distinguish (A1) from (A2). For Strawson meaning is a function of the sentence (A1), whereas "mentioning and referring and truth and falsity"  are functions of the use of the sentence (A2). Thus to talk about the meaning of a sentence is not the same as talking about its particular use on particular occasion, but about "the rules, habits, conventions governing its correct use, on all occasions, to refer or to assert." In the case of the sentence S: "The King of France is wise", Russell's problem was to explain how the utterance of a sentence that contains a non-existent term can nevertheless say something and have a truth-value. Strawson's answer is to employ his distinction between meaning and mentioning, (A1) and (A2). In other words, the thing I mean when I use an expression "is quite different from the meaning of the expression I use to talk of it."  An example is the word "this". If someone asks me it's meaning I do not point out or hand them everything which the word mentions or to which it refers. Instead I explain the general rules which govern its uses in particular utterances. What is important to grasp, Strawson argues, is that the significance of this sentence is independent of the potential truth-value of a particular use of it. It is Russell's failure to grasp this that leads to "the troublesome mythology of the logically proper name." Stawson denies the validity of Russell's logical reduction. He disagrees that the utterance of S logically implies the existence of D: "The King of France", and he argues that when we respond to such an utterance by saying "There is no King of France" we cannot be said to be directly contradicting S. Rather, according to Strawson, we are "giving a reason for saying that the question of whether it is true or false simply does not arise." In other words, to begin a sentence with the expression "the so-and-so" implies (in the sense of signalling) that the existential conditions (described by Russell) of the particular individual mentioned are fulfilled but it does not state that they are. Russell fails, in Strawson's opinion, to make this subtle but necessary distinction between a disguised assertion and a mere signal.
Once again we return to Strawson's central point: that just because a sentence is significant does not mean that any particular use of it must have a truth-value. This is close to Frege's position, but considerably more sophisticated because it is not based on the literal identification of the truth-value as the references of sentences, but rather on the careful distinction between a sentence as type, its use and its utterance. If a sentence qua sentence is significant that just means it could be used, in certain circumstances, to say something true or false. It need not necessarily always be so used.