Popper's epistemology emphasizes the importance of subjecting our scientific theories to rigorous tests in an effort to falsify them. We know that a theory is false if an experiment shows that one of its predictions is false. At least, that's the simple version of the story.
The Duhem-Quine thesis, which is often presented as a criticism of Popper's epistemology, points out that when we perform an experiment to test some theory T, and the experiment seems to show that one of T's predictions is false, it does not necessarily follow that T is false. Whenever we try to test one theory, we always make use of many other theories. Popper calls these other theories "background knowledge". So when the results of our experiment seem to show that T's prediction is false, we don't necessarily know that T is wrong. Instead, it might be that one of our background theories is wrong.
Does this point mean that Popper's epistemology doesn't work? Thankfully not, and in fact, when properly understood, Popperian epistemology already accounts for this.
While an experiment can't tell us that the particular theory we intended to test is wrong, it can tell us that at least one of the theories involved in the experiment is wrong. When we say that a theory's prediction has been falsified, what we really mean is that that one of the consequences that the theory entails has contradicted experimental data - or more accurately, our interpretation of the experimental data. Of course, what consequences a theory entails depends on what other theories we have available, and the process of interpreting experimental data is itself a process deriving consequences from theories, in the context of that data. So ultimately what a falsifying experiment shows us is that a set of theories - all those involved in deriving the prediction or in deriving the interpretation of the data that the prediction conflicts with - entails a contradiction. And when we notice that our theories entail a contradiction, we must seek to get rid of (or modify) at least one of those theories, in order to resolve the contradiction (as, according to Popper, the commitment to resolving contradictions is an essential driving factor behind intellectual progress).
Once we know that a set of theories entails a contradiction, and that we must therefore reject at least one of them, we're left with the question: which theory should we reject? If we were to adopt the naive reading of Popper that the Duhem-Quine thesis is often intended to refute, we might say: always reject the theory that the experiment was intended to test. But, of course, there's no logical justification for that approach, and a careful reading of Popper reveals that he never advocated for that answer. The true answer, I think, is slightly more complicated.
At any given time, we have some set of things that we wish to explain, and if we're lucky, our current theories may in fact allow us to explain some of these things. However, if we were to discard a theory, we may lose our ability to explain some of the things we wish to explain - we may lose some explanatory power. When deciding which theory involved in entailing a contradiction we should reject, we should examine each theory and ask ourselves: if we rejected this theory, would we lose the ability to explain any of the things that we wish to explain? In other words, would rejecting this theory mean losing some explanatory power? If not, we can reject the theory, but if so, we must not reject it. The answer to the question "which theory should we reject?" is therefore "whichever one can be rejected without losing explanatory power". This may be the theory that our experiment was intended to test, or it may be some background theory - it depends entirely on the nature of the theories themselves, and on what kinds of things we wish to explain.
However, we may sometimes find that none of the theories involved in a contradiction can be rejected without losing explanatory power. In this case, we have no acceptable way of resolving the contradiction, and we have no choice but to tentatively hold on to all of our theories, even though they entail a contradiction. So sometimes the answer to "which theory should we reject?" is "None of them, yet". This is precisely the situation that the physics community is in now with regard to the contradiction between quantum mechanics and general relativity.
When we find ourselves in this situation, where we have some contradiction that we can't resolve because all of the theories involved are necessary to explain all of the things we wish to explain, our only choice is to try to conjecture new theories. If we're lucky, we may manage to find a new theory that can replace one of the contradictory theories and allow us to resolve the contradiction. Specifically, we need to find a theory that can allow us, when we adopt it in place of some other theory, to explain everything that we wish to explain without entailing the contradiction. In the aforementioned case of quantum mechanics and general relativity, we would need to come up with a theory of quantum gravity that can replace one or both of them: a theory that explains everything that we want to explain about gravity and quantum phenomena, without entailing a contradiction.