This is Popper's early masterpiece, which still deserves to be thoroughly read. Thesis of the book: theories are guesses which have no secure basis and can be at any time overthrown, but which must be able to stick out their necks and face experimental tests. If they pass the tests, this does not make them any more secure or reliable than they were before.
Its first chapter explains two fundamental problems which will be grappled with in the following chapters: the problem of induction and the problem of demarcation (between science and non-science). The solution to the first problem is straightforward: there is no such thing as induction. If you want to learn more on Popper's formulation and purported solution of this problem, you should read the whole book.
The second chapter gives some methodological rules which, though presented as conventions, are set down in order to combat "conventionalism", the attempt to regard theories as irrefutable, as true by convention.
The third chapter, a bit boring, is an analysis of causality, scientific explanation, the kinds of scientific concepts and the structure of theories (these are considered interpreted axiomatic systems).
The fourth chapter deals with the notion of falsifiability, something theories must have in order to be scientific according to Popper's criterion of demarcation. Falsifiability, as here defined, is (roughly) incompatibility with at least one singular statement reporting the existence of an observable event (the distinction between occurrences and events will be found here; it was previously drawn by Bertrand Russell, I may add).
The fifth chapter deals with these last kind of statements (basic statements): their form, their content and their role in science. These statements are in no sense justified by experience, says Popper, even if their acceptance is caused by experience; they are as risky as theories, although in scientific practice there is not (usually) much trouble in agreeing to accept or to reject them. It is a pity that Popper says that basic statements are accepted by a "free choice" or convention, because it is only after observing that the popperian Forscher will agree to accept a basic statement.
The sixth chapter tries to define comparative criteria of falsifiability. Given that all scientific theories have an infinity of observable consequences, how are we to compare their boldness = refutability = their sticking out their necks?
I am running out of words. The seventh chapter deals with the notion of simplicity. Popper's thesis here is: simplicity = boldness = falsifiability; a simple thesis, and a bold one.
The eighth chapter contains a deft and clear discussion of some methodological and mathematical problems of probability. I highly recommend it. It is after reading a chapter like these that you can realize how cheap and misleading the criticisms of Stove are to which some previous reviewers refered.
Chapter 9 contains a plea for objectivism in quantum physics, although it is rather out-dated. But the attack on Heisenberg's programme is still instructive.
The last chapter deals with "corroboration" of theories and includes an important critique of justificationist probabilism. One should read it together with Reichenbach's highly negative Erkenntnis review: "Über Induktion und Wahrscheinlichkeit: Bemerkungen zu K.Popper's Logik der Forschung".
The appendices are also worth reading, even if they tackle mainly with technical problems.
I think that no one has seen with greater clarity the problems and ambiguities of Popper's methodology as displayed in this book than his coleague-rival Imre Lakatos. Even if he is not half as gifted as Popper, and makes many mistakes as regards induction, his critique of popperian demarcation and rules of science is certainly worth reading.
On this book, one can also benefit and enjoy reading Neurath's indignant review of the 1934 edition: "Pseudo-Rationalismus der Falsifikation", and Grelling's review in "Theoria", 1937 (1).