Now I’m talking about hard scientists here. Astronomers and Physicists and anyone else who’s involved in studying atoms/quarks/black holes/anti-matter, etc. You get the idea. Guys like Einstein, Fermi, Feynman, etc. Just what the hell are they looking for? You often hear words thrown around like “find the key to the universe”, or “solve the mystery of life”, or some other such meaningless phrases. So, what are they looking for and what are they going to do when they find it?
In my admittedly limited experience of scientists, they’re looking for the explanation for something.
And after they find that, they’ll start looking for the explanation for the explanation.
Fermi worked on the problem of nuclear fission and so contributed to us getting nuclear weapons and nuclear power plants. Einstein’s theory of relativity helped to explain some observations in astronomy and, more interestingly, is actually used in the workings of GPS. What was the point of what Feynman was doing (beyond making a salutary effort to explain physics in a more understandable way to college kids) is utterly beyond me.
The above suggests that these scientists tinker with mathematical formulas, sometimes build complicated apparatus to run physical experiments based on the formulas, and sometimes these formulas of theirs end up being used in making exciting gadgets that destroy cities or help us drive from point A to point B.
But that’s just the old school physicists, though. Reputedly the new-fangled “string theorists” prefer to play around with math formulas that are not expected to be verifiable or useful any time soon.
Generally, they’re looking for something nobody else has seen or noticed.
Feynman was involved in quantum mechanics (among other things) and he is considered as one of the fathers of quantum computing and of Nanotechnology.
They are curious about the strange and interesting world they find themselves in, and are curious to better understand the rules of the universe. The general thinking is that the rules are not random – there is some interesting and perhaps ‘deep’ reason for them. It’s one of the greatest and most perplexing mysteries, and it seems natural that a curious individual would want to try to solve it.
Phrases like “find the key to the universe” are designed to sound like they mean something, but they don’t. I dislike them.
The scientists have figured out various things that fit together and sometimes predict other surprising things we can go out and verify. This is intoxicatingly interesting to the people involved, and keeps creating information that society values and uses, so everybody wins.
I think the “what” scientists are looking for is whatever interests them, and that’s just fine.
Generally speaking, scientists are trying to solve rather specific problems that arise from the science that has been done before, when it fails to account satisfactorily for particular phenomena that it seemingly ought to be able to explain. Unfortunately, unless someone already has an intimate knowledge of that previous science, it is usually very difficult to explain to them why a current scientific problem is interesting or important. That is why they often resort to blather about “the key to the universe” or whatever.
It is also why reports of new scientific findings often include wildly speculative claims about possible relevance to practical issues (curing cancer, or whatever), most of which never seem to pan out. Experience shows that increasing scientific knowledge does, very often, turn out to contribute towards the solution of important practical problems, but (in the sort of ‘pure science’ you are asking about) that relevance is usually much more indirect than a particular discovery leading directly to a solution to a particular practical problem. Rather, the more fully we understand a relevant aspect of reality (e.g., the complex factors that regulate the growth of cells in the body) the more likely it becomes that we are going to be able to figure out some practical way to control some aspect of it (e.g., to prevent the uncontrolled growth of cells that is cancer).
A group of people observe X.
Some of them ask “Why X?” and begin proposing theories (much debate ensues).
Some of them try to take apart X to find out how X works.
Some of them try to replicate X in a controlled environment.
Some of them adapt X into X’ to suit new and specific purposes.
Some of them ask “What’s after X?”.
(Have I left anything out? I probably have - feel free to add).
There are many unsolved problems in physics. Once something is understood, it may be possible to find practical applications for that knowledge. It’s rarely possible to know how useful a particular discovery will be before it is made. For example, building transistors requires an understanding of quantum mechanics, an invention that made the information age possible. The ultimate goal is to find a unified theory of everything, capable in principle of predicting the outcome of any experiment.
However, I don’t think practical benefits are necessarily required to justify scientific research. Understanding is itself a worthwhile goal.
I dislike this kind of guff as well. I’d just say, to understand as much as we can understand.
they are looking for a conception of reality. some of the time it is new conception reality or a better conception of older reality.
An essential measurable observable definition for the term “mass” would be a good thing to find. A way to say “here are x number of particles and here’s how they all join together and add up to be an ounce of some actual matter” - that would be kindof cool and there might be a few scientific applications to having that kind understanding of subatomic physics.
They are looking for the Grand Unifying Theory that ties everything together. All the forces at the macro and micro level.
Ultimately, scientists are looking for a large research grant 
I think the most intriguing thing about the Hadron Collider is not what they hoped to find (whatever that is), but how the eggheads convinced the beancounters that the results would be worth the investment.
Put simply they are doing what man has been doing ever since he was man, and probably his primate ancestors before him - breaking things apart to see what’s in them. Sure, the methods now are more sophisticated and we use accellerators to hurl particles together with the force of a million suns, but it’s still the same process.
“We gave you nuclear weapons - the power to destroy an enemy completely, or the whole world - just a few decades ago. Give us what we want and just see what we give you this time!”
Can we finally just replace http://boards.straightdope.com with a redirect to Wikipedia??! 
String theorists and others looking for end theories are a percent of a percent of working physicists and therefore another percent less of all working scientists.
Most working scientists have specific research projects in their lives rather than theoretical exercises. (Theoretical physics papers have one or two authors; experimental physics papers have one or two hundred.) I’m pretty sure more scientists work for industry than for academia, as well. But their lives are the same. They search for the next answer, the next piece of the puzzle, the next breakthrough. You hear more about the mark the Big Thinkers make in history, but that’s like comparing all the hundreds of thousands of politicians with the handful of Presidents. It’s an almost worthless comparison.
I prefer “tear the mask off nature and stare at the face of God” myself. It has more panache and publishers eat it up.