Is there any other way to describe work done instead of saying the force applied to move an object through a distance? I was reading about energy, and I got to know that energy is the ability to do work. Then I read about work, and I found that almost everywhere, resources say that when an object applies force to move another object through a distance, it is said to do work, and the Math is that $W = Fs$(Even the tag on this site which says work). So my first question is:

Can work be described in another way instead of saying this? This is
  not making a complete intuitive sense to me. And, why is it defined
  like this? What is the reason behind defining something called work?
  Is there any derivation to it?

And I am also not able to understand negative work. Lets recount the example of friction, which opposes the motion of an object, so we say that the work done is negative. But, if I go by the definition, then I think that friction is not making the object move through a certain distance, it is trying to stop it. 

So what role does work play here?

And I saw many physics problems which looked extremely difficult to solve, but with this concept of conservation of energy, they were looking very simple. But still, I am not able to understand the reason behind the formulation. 
Is this formulation more of an empirical result, or is it something else that I am missing?
I am not opposing this, just asking logic behind things. I hope readers are not annoyed with this :)
 A: To back up CuriousOne. A good example would be you sumo wrestling with your friend. Lets say your friend is far bigger/stronger than you. So you begin pushing one another. His force on you is greater and so you are obviously moving in what you would call your -x direction. Both of you move in the -x direction, but of course your exerting a force on him too in your +x direction. Both of your are moving in the -x direction, from your point of view (your x-axis), his greater work is -ve (opposing where you want to go) while your work on him is positive. If you push with less and less force, he would travel further so you were impeding him (doing negative work with respect to his x-axis). If you pushed each other with the same force, you'd both be still so no work done. 
A: In another way, we can define work as "the change of energy".
When you do work on a system, you change its energy. So positive means you're increasing the system's energy (like pushing a trolley forward), and negative means you're decreasing it. 
In this context, it might be easier to see why some people are inclined to consider friction as doing "negative work", because it mostly takes away the energy from the system (like slowing down a trolley). (But this is not always true, so I for one am not really fond of this distinction.)
A: Quantity that we call work defined as scalar product of the segment of a path and the force applied equals the quantity which we call kinetic energy of an object and/or potential energy of an object. These are all very exact quantities. Furthermore, energy is good for calculations of a motion or predictions of future changes of some variables of a system. So these are all very usefull in calculations. Work can rewritten in some other way, I dont know, but it is the quantity with units of force times units of distance that plays big and important role in physics. So, if you are asking if there is some other mathematicly good and intuitivly usefull quantity that has some other units and stil can in some way be called work, I would say no. In modern physics concepts like Hamiltnian and Lagrangian and thir respective equations are of great use and Hamiltonians or Lagrangians all are defined through kinetic and potential energy of th system. I hope this sheds some light.
