I suppose the title may be a white lie - I actually want to talk about Thomas Kuhn's approach first, and then I can get on to his account of normal science. See, prior to The Structure of Scientific Revolutions (hereafter referred to as Structure), Logical Empiricism held sway for most discussions of the philosophy of science. They're a very interesting bunch, and in fact I'll no doubt cover them someday, although I'd like to get my hands on some primary literature by A. J. Ayers and Carl Hempel before I try to discuss it much. Basically, the Logical Empiricists (also known as Logical Positivists, although they didn't like the association with Auguste Compte from being called Positivists) were very concerned with working out the normative description of science, and in giving it a special epistemic status wanted to work out the logic governing "good" scientific practice.
Kuhn would have none of it. Well, maybe that's overstating, but he didn't see eye to eye with them. His original training was in physics, but he was instructed to teach a course of the history of science for non-science students, and the rest, as they say, is history. As opposed to a logic of science, or any progressive pattern, he saw a number of seemingly rational yet incompatible theories arising in conjunction. If there was to be something epistemically special about science, it had to be expressed in science as practiced, not science as constrained and never practiced.
So he bases his story of science and its special status by examining its history, and here we have a rather novel (for its time) approach. There are two big parts to Structure - normal science, and revolutionary science.
Here, I'm primarily concerning myself with normal science. This is partly because it's treated by Kuhn first, and partly because understanding revolutionary science doesn't make much sense without understanding normal science first. I'll get to revolutionary science later, and then go back and analyze it a little more in depth after that. However, before any critical treatment of the arguments, I like to take the writing for granted until it's been digested.
So we have normal science - in Kuhn's description, this is "researched firmly based upon one or more past scientific achievements, achievements that some particular scientific community acknowledges for a time as supplying the foundation for its further practice." These achievements are "sufficiently unprecedented to attract an enduring group of adherents away from competing modes of scientific activity" and "sufficiently open-ended to leave all sorts of problems for the redefined group of practitioners to resolve." (If I quote Kuhn heavily, it's because he's actually very readable.) Achievements of this sort result in a research program as governed by a paradigm. (If you've heard that word used in common usage often, blame Kuhn.)
A paradigm for Kuhn is basically the school of thought, complete with theories, heuristics, and instrumentation, that arise from a successful and sufficiently convincing achievement. The whole idea will go together a little better, regarding how paradigms arise, once we get to revolutionary science. For the mean time, suffice for a rougher understanding of a paradigm.
A paradigm establishes itself and researchers run with it. You have your overarching paradigm, which includes a set of theories, like the Newtonian laws of motion in the Principia, as well as your examples of problem solving within the paradigm. New students to the science learn about relevant examples in the paradigm by training, and are generally indoctrinated into it, being given examples of application. These are generally not explicit, else there would be little use for the hammering out of details that governs normal science - in fact, this non-explicit statement leads Kuhn to pull out the Wittgensteinian "game" analogy, noting that learning is by iteration of similar things with the goal of getting a feeling for the methods acceptable to the paradigm.
So within the paradigm, the benefit of normal science is puzzle solving ability. Kuhn describes this as puzzle solving because researchers tackle problems for which there is presumed a solution. This solution is generally expected by the theory but not confirmed. He notes explicitly that paradigm governed research is, in fact, a lot of "mopping up" of the details, tedious work that is generally not acknowledged by non-scientists. Paradigms don't aim for "unexpected novelty." (A break with Popper, certainly, who held that disconfirmation and novelty was the business of science. We'll get back to him in revolutionary science, though.) Sometimes anomalies are found, especially in paradigms that encompass a sufficiently large aspect of natural inquiry, and in these cases either the theories are tweaked or refined. However, it is generally not accepted within most paradigms to just 'add in' and explicit statement of '[Y holds] except in instance of [anomaly X]'.
Part of the puzzle solving ability also lies within the apparatus used by scientists. Kuhn makes a strong point of early research in electricity attempting inquiry by use of a pan balance, and found no consistent regularity - it was not until Coulomb that special apparatus was created for measuring point charges, and these became objects of inquiry once the means of measuring them was available. If there are questions which are not accessible by the instruments accepted by the paradigm, then the research program encompassed by the paradigm generally ignores them.
While paradigms usually don't do much in the way of novel discovery, Kuhn does seem to have a role for discovery and invention (terms he uses almost interchangeably, since there seems to be little difference for him in stating that one discovers facts or that one manufactures facts - for reasons to be evident later). He notes in particular the discovery of X-rays. Although the paradigm of prior established electromagnetic radiation explicitly allowed for X-rays, practitioners were dismayed to find that cathode ray tubes were emitting more than they thought they were - and so research had to be re-done to establish that it was not confounded. This, for Kuhn, still falls in the heading of normal science, since the paradigm wasn't replaced with anything new, but simply tweaked to make room for a new phenomena that fit beside what was already established. In this means, Kuhn argues, there can be novel discovery without a paradigm shift.
Okay, we've covered half of Kuhn now. I know that's kind of long, but HALFWAY THERE. And we're past the boring part. Revolutionary science is where all the excitement is!