While technically a late-nineteenth century mathematician, Frege is usually lumped together with the early Analytic philosophers of the twentieth century, including Russell, Moore, and Carnap. This is mostly because Frege's work was not taken very seriously by the European intelligentsia of his own generation; in Carnap's autobiography (Carnap was a generation younger than Frege), he describes taking a course on formal logic from Frege with just two or three other students.
Über Sinn und Bedeutung presents Frege's basic theory of semantics. The best translation of the two critical terms -- Sinn and Bedeutung -- is an issue of some controversy among Frege scholars, and for the sake of charity I will follow the practice of using the untranslated terms here. Of the two, Bedeutung is the slightly more straightforward notion: the Bedeutung of a term is the object to which it refers, its denotation. The Sinn of a term is related to 'its mode of presentation', but is still objective and interpersonal. This is probably best illustrated with Frege's own example of 'Venus', 'the morning star', and 'der Morgenstern'. The Bedeutung of each of these terms is the same -- the particular planet. However, 'Venus' has a different Sinn from both 'the morning star' and 'der Morgenstern', while the latter two have the same Sinn. Furthermore, 'the morning star' and 'der Morgenstern' are distinct as signs -- one is English, the other German -- and each will be related to a different Begriff [concept] in the mind of each speaker who uses these terms. Thus, we can arrange this system of references in order of decreasing subjectivity:
Frege's primary concern is to sort out just what the Sinn and Bedeutung of sentences are. His reasons are logico-epistemological: How do judgements of the form 'a=b' work? Do they assert a relationship between mere signs, or between objects? If between mere signs, then, without the theory of Sinn and Bedeutung, there's nothing objective -- literally, no objects -- for the judgement to latch on to; but if between objects, 'a=b' is always equivalent to 'a=a', so that all truths would be analytic or logical truths. Frege's solution is that 'a=a' and 'a=b' have the same Bedeutung, but different Sinne. In particular, under normal circumstances, the Bedeutung of any sentence (if it has one) is its truth-value, either The True or The False. Hence, the Bedeutung of any sentence of the form 'a=a' is The True, as is that of true judgements of the form 'a=b'. It is the Sinne of these sentences that serves to differentiate them, and thereby differentiate the assertions a=a and a=b; while their objective truth-values ground the judgements as knowledge.
Russell, On denoting
Russell was British, and perhaps the most prominent of the early Analytic philosophers. On denotation presents a semantic theory which he contrasts with Frege's of Über Sinn und Bedeutung, though I don't understand the criticism (it seems to depend on identifying expressions with their Sinn, and many of what Russell seems to think of as great improvements over Frege can actually be found right in Frege). The only real difference between the two is that Russell distinguishes sharply between proper names, eg, Scott, and denotating expressions, eg, the author of Waverley. For example, consider the sentence 'Scott is the author of Waverley'. Frege's analysis would go as follows:
Let s denote Scott and w denote the author of Waverley. Then we paraphrase the sentence as 's=w'. The Bedeutung of this sentence is The True.
Russell's is more complex; 'the author of Waverley' is paraphrased into a conjunction of several predicate forms:
Let S be the predicate 'is Scott' and W the predicate 'wrote Waverley'. Then we paraphrase the sentence as (Ex)(Sx & Wx & (y)(Wy -> y=x)).
Note that one immediate consequence of Russell's analysis is (Ex)(Wx), ie, 'There exists someone who is the author of Waverley'. Russell believes his analysis works in the important case of non-existent entities, while Frege's does not. For example, take the sentences
(1) The present King of France is bald.
(2) The present King of France is not bald.
The former is analysed into something of the form (Ex)(Kx & Bx), and is false because it implies (Ex)(Kx), which itself is false. But the latter is ambiguous, and can be analysed into either of the following:
(2a) (Ex)(Kx & ~Bx);
(2b) ~(Ex)(Kx & Bx).
(2a) is false (because it implies (Ex)(Kx), which is false), while (2b) is true. Therefore, (2) has no truth value per se.
Interesting philosophical link: What actually IS 'design'?