Popper popularized the notion that empirical science can not be verified, only falsified. Isn't he right? How can someone verify a theory? To me, it does not seem possible. However, you can easily falsify a theory. I've searched the web for some answers and have not come across anything that I find satisfying. It seems like people don't like the fact that we should perform 'science' based on falsification. Can someone clarify the issues with Popper's theory?
Someone can verify a theory by working out its predictions and testing if they are confirmed. This approach was developed by logical positivists, especially Carnap, under the name "verificationism". Unfortunately, they never came up with a quantitative "degree of confirmation", and their philosophy of distinguishing theoretical and factual truths, required for verificationism, did not work out either. Aside from that, Popper's main objection was that one can not test all (infinitely many) predictions of a theory, and finitely many confirmations are not definitive and prone to bias, one might be inclined to look for predictions that will be confirmed. Popper's solution is to look for predictions that are "least likely" to be confirmed, and actively attempt to falsify a theory. Failure to falsify despite the best effort serves as endorsement of the theory.
Although falsificationism is "morally right", we should test our theories against our experience, and judge them based on that, it is too simplistic to work. Its problems are not of the sort that can be solved by turning back to verificationism however. As Popper himself concluded the theory of evolution is not a scientific theory, according to falsificationism, but a metaphysical programme. Similar issues arise in geology, sociology, linguistics, psychology, economics, etc. In fact, the further a discipline is from physics, the less progress in it can be explained by falsifications.
One problem is that falsificationism does not account for where the hypotheses to be tested are coming from, they just "appear", and then are kept or discarded. Especially outside of physics a hypothesis with explanatory value over a range of instances, which is falsified, is still valuable enough to be kept around rather then discarded. It may serve as a substrate for further hypothesizing, or even just for the sake of coverage it does provide, especially if there is no clear alternative. Contrary to falsificationism, this puts value on verification of confirming instances, one is more interested in the scope of confirmations than in existence of falsifying anomalies (assuming they are rare). It also means that we need means of choosing between theories that are all partially confirmed and partially falsified based on grounds that are largely non-empirical (explanatory value, coherence, economy of means, utility, "elegance", etc.), and falsification is not very useful in this regard.
Another problem is that precision of theories sharply declines as one gets outside of physics, reflecting the decline in exact laws that govern those areas. Psychoanalysis in psychology, evolution in biology or universal grammar in linguistics are not up to the exactness standards of relativity or quantum mechanics. They are too vague to be outright falsifiable in Popper's sense, but trying to make them more precise may not be fruitful, because regularities that do exist are themselves devoid of such precision. In other words, we might be discarding what is there because it can not be expressed in a strictly falsifiable form, while looking in vain for what is not there. Kuhn's analysis of the history of physics shows that even there progress is often based on procedures that would be proscribed under falsificationism, such as "protective belt" of extra hypotheses to save a theory, which is allowed by diluted falsificationism of Lakatos.
You should read David Stove Popper and After: Four Modern Irrationalists for a detailed analysis. An annoying meta-problem with Popper's claim is that it doesn't apply to all scientific theories, it only applies to theories with unbounded domains. (His reasoning depends on the impossibility of inspecting an infinite set). A concrete problem pertains to the idea of a "falsifying" observation. If a theory predicts that a certain object is supposed to exist at a particular time and place and that object is not observed then and there, we typically presume that the theory has been falsified, though we may want to make sufficient numbers of observations to rule out the possibility that we blinked at the crucial moment. Assuming that the failed observations are repeatable, we would also require evidence that the observation itself is valid, meaning that there is a true and verified theory underlying the theory of the instrument. But according to Popper, one cannot versify that the instrument shows what it is believed to show, since that rests on a theory that we have either shown to be invalid (false) false, or else that we have not shown to be false. Failing to show that something is false does not constitute proof that it is true. And so the falsifying observations have the power of being falsifiers only if the underlying theory is true, which cannot be determined.