This summary follows a systematic slightly different from Popper's, whose demarcation and theory selection criteria are a little more intermixed and spread  with different wordings, over  various chapters and books. The references given here are just some examples.

Demarcation

With the demarcation of scientific theories from pseudo-sciences Popper aims to disqualify scientific claims not only by astrology but also of speculative theories as those of Marx, and Freud. He stated that scientific statements have to comply with the following requirements:

1. It shall be Logically consistent and not be a tautology;[Popper 1959/1972: p32] because a tautology does not add knowledge although it might give the impression to do so;
   
2. It should be 'inter-subjectively' testable; .. 'a scientifically significant physical effect may be defined as that which can be regularly reproduced by anyone who carries out the appropriate experiment in the way prescribed' [Popper 1959/1972: p44, 45 and Popper 1963/1972:p257:] Although Popper avoids the word, to distinguish his approach from the Wiener Kreis, this could be seen as 'verifiable'.
   

It should be possible to falsify the statement;[Popper 1959/1972: p40] This requirement should be understood in a methodological way, this means even when it is not possible to falsify the statement yet , it should in principle be possible. Anyone who makes a scientific statement should indicate by which empirical evidence he would withdraw his statement. This criterion is Popper's most important one and the one he uses to distinguish his approach from the logic positivists.

Preferred Theory

As long as a theory corroborated by withstanding all kind of sever efforts to falsify that theory there are good reasons to continue using the theory.
A next step should be made in case there are two competing theories. Popper defined the following set of criteria to make a choice between two (or more) theories:

1. 1.A new theory that constitutes a scientific advance is preferred, for instance it includes the older one, and the extension of the newer one is supported by empirical evidence.[Popper 1959/1972: p33]
   
2. Information content: The theory that offers more relevant predictions should be preferred. And as a consequence in general such theory would expose itself more to falsification.[Popper 1963/1972: p256]
   
3. The theory which is easier to verify is preferred. [Popper 1963/1973: p386]
   
4. When otherwise equivalent, the most clear and simple theories should be preferred.[Popper 1963]