Popper's central claim is that a statement is scientific only if it is falsifiable. Note that this condition is necessary but not sufficient: a statement can, for example, be falsifiable and actually false, in which case it is not scientific. (Some might argue that Popper never says this, but it seems so trivial that if he really doesn't say so explicitly, it is uncharitable to believe he doesn't take it for granted.)
A statement is falsifiable if and only if there is an empirical observation that could in principle actually be observed that would render the statement false. If we do in fact observe a falsifying observation, then the statement is definitely false. However, if we bend over backwards to attempt to observe something that would falsify the statement, and we are unable to do so, we gain confidence in the truth of the statement; we can use probability theory (Bayesian or frequentist) to quantify our confidence. We can be certain that some statements are false, because we make a falsifying observation, but we can never be certain that any statement is actually true.
For example, take the statement, "Objects always fall when we drop them." This statement is falsifiable: if I drop something, and I observe it not falling, then the statement is definitely false. However, no matter how many times I drop something and observe it actually falling, I cannot be certain that things always fall when I drop them, everywhere on Earth for all time. The best I can do is say having observed things falling when I drop them many times, I have confidence in the statement, but if I were to ever actually observe a counterexample, then I would have to revise my theory.
In contrast, consider the statement, "God created the universe." The syllogism
- If God had not created it, the universe would not exist
- The universe exists
- Therefore, God created the universe
In principle, Popper's claim is very straightforward; in practice it's a lot more complicated. Most importantly, we cannot evaluate statements independently: all statements depend on a theoretical context, the larger structure of a scientific theory, the construction of our measurement devices, and the nature of language itself. Hence, we cannot isolate single statements for falsification; instead we can only falsify larger theoretical constructs. The best we can say if we observe something that falsifies a statement within a theoretical context, then something in the context (including the statement) is false, but we can't be sure precisely what part of the context (or the entire context) is incorrect.
For example, I'm watching the Feynman lectures. Feynman talks about observations of the moons of Jupiter that falsified Newton's theory of gravity. Instead of modifying the theory of gravity itself, scientists changed the theoretical context by adding the premise that the speed of light is finite, with a definite value.
But this ambiguity does not damage Popper's central point. Popper does not give us any guidance on how to fix problems, he merely claims that if it is impossible in principle to have this particular problem, i.e. impossible to observe something contrary to the theoretical context, then whatever it is you're doing, you're not doing science.