You asked me to simplify what I wrote about Popper, Kuhn, and Lakatos. Here's another version of what I wrote - maybe it will be smoother and less technical. I am not sure that it will be helpful because others, notably Riddikulus and Mina, made great points that contribute into this discussion a lot better than mine. But I'll still try. Again, the whole point here is that there is no such thing as one universal scientific method.
Early scientists were mostly positivists and "inductivists," meaning that they thought that science begins with factual observations ("unlike non-science, science is derived from facts"), and that conclusions in science are made by induction - if something is true in this situation and in that and in that, then this something is true in all situations. As you yourself correctly stated, Popper challenged this approach, based on two reasons: 1. Factual observations are still "theory-laden," i.e. even to observe something, you must already have a concept, a theory, a framework (Kantian "category") in your mind, and 2. Induction is, strictly speaking, wrong from the point of view of the formal logic. Popper attempted to replace the inductivist approach to science with a "falsificationist" approach, which means that actual science may begin not from observations, but from a rather voluntaristic statement that may not, per se, correspond to factual observations. However, this statement must be falsifiable, meaning that if evidence is found that there exists something incompatible with the made statement, or with logical deductive predictions that follow from this statement, it must be withdrawn. If my hypothesis is that all swans are white, and then suddenly someone finds just one black swan, I take my hypothesis back and begin to think about another hypothesis.
While Popper's ideas are universally recognized as interesting, some practical observations contradict them. Duhem in the 19th century, and Quine in the 1950-s wrote that scientists actually do not always wish to take back their hypotheses when they seem to be falsified by others. For example, the hypothesis that the Earth moves could be "falsified" in the 16th and 17th century because of the so-called tower phenomenon. Throw a stone from the top of the tower, and it will land near its foot - hence, the Earth does NOT move. However, scientists like Galileo objected to that, saying that in fact the stone moves forward together with the moving Earth (implying what we call an "auxillary hypothesis" of inertia). Another example may be that when medical researchers found that stress, high blood pressure, high blood sugar etc. can cause certain diseases, this seemed to "falsify" Robert Koch's hypothesis that every human disease is caused by a microorganism or an infectious agent. Yet, proponents of the Kochian idea said, well, it's just that our methods, technologies are imperfect, so we fail to see the infectious agent, but it is there. And in many cases they turned out to be correct: for example, we now know that Helicobacter pylori is the cause of peptic ulcer and (more recently) a retrovirus is the cause of the chronic fatigue syndrom. So, falsification is not a panacea either, it does not seem to really work as science progresses.
Challenging Popper, Thomas S. Kuhn developed a theory of "paradigm shift." According to Kuhn, science develops in cycles. It always begins with a revolutionary - visionary person stating something that is accepted as a "paradigm" (central notion): for example, that unless a body is acted upon by some forces, it keeps moving without acceleration or deceleration. (That's the famous first law of Newtonian mechanics: note that it was not a record of any factual observation and it could be "falsified" VERY easily at the time of Newton). Then comes a period of what Kuhn called "regular science." It means that a growing number of scientists begin to develop the paradigm, looking at various situations where this paradigm can be applied. They do not produce any new paradigms and, moreover, they develop a "tunnel vision" in that they refuse even to recognize anything that does not fit the paradigm they re developing. That leads to stagnation and crisis. Then, during the crisis, a next visionary enters the scene and announces a totally different paradigm - and the cycle repeats.
Imre Lakatos was not satisfied by what he viewed as "subjectivism" in Kuhn's paradigm shift theory, and tried to develop his own theory of science, known as the theory of research programs. According to Lakatos, science is, indeed, moved forward by paradigms. Yet, these paradigms do not change each other arbitrarily. Rather, a paradigm, or a "hard core" statement, gives rise to a "research program," i.e. a network of people working on this paradigm (again, much like "normal scientists" in Kuhn's account), developing its "protective belt" - a large and growing number of auxillary statements, hypotheses, theories, etc. Several "research programs" work in parallel, and some of them become "progressing," i.e. their "protective belt" is being challenged, partially falsified, and changing, while their "hard core" (which, importantly, gives rise to the protective belt) remains the same.
I do not know Karl Hempel's works, - thank you for pointing out that they exist; I will certainly lok into that. However, I hope the examples above illustrare the idea that the mere notion that there is, or even should be, some unique, well-defined, "one-size-fits-it-all" "scientific method" is simply unsustainable. We do not quite know how science develops, why it makes progress. Diferent philosophers explain this quite differently. Studies of some very famous scientists' personal journals indicate that their methods (in the philosophical sense) differed greatly, and that they, in their pursuit, often violated the neat "scholarly" rules of "THE" scientific method.
And again, most importantly: yes, NOTHING in science is ever "proven." And yet, science exists and makes progress and continues to serve the humankind.