Force is not really a fundamental concept in physics, although students are often introduced to physics through the forces. More fundamental concepts are momentum and energy.
A force is that which changes the momentum of an object or particle, or changes the shape of an object. The force equation is the best definition; Newton's First and Second Laws (as stated in plain language in many beginning textbooks) offers implicit definitions of a force:
"An object in motion tends to remain in motion at a constant velocity, and an object at rest tends to remain at rest, unless acted upon by a force."
"The acceleration of an object whose motion is unrestricted is directly proportional to the force on that object, and inversely proportional to that object's mass."
Now, that said, we must also remember that a force applied to an immobile but flexible object can deform that object. The equations that describe this are far more complex; and because the object does not move, not only can we not measure the force in terms of the acceleration, but also (obviously) the object's mass does not play a part. Thus, we must treat the force as if it were making some object move in order to understand it.
A force applied over a distance is energy. And energy applied over time is work. So a force applied over a certain distance, in a certain period of time, is also work.
There are four forces in nature: gravity, electromagnetism (this looks like two forces, but is actually one), the weak nuclear force, and the color force. What we often mean when we speak of a force, that is, contact between two objects, is actually the electromagnetic force in disguise.
Its disguise is called a van der Waals force, named for the Dutch physicist who first described it. What happens is that the electrons in the atoms of one object get close to the electrons in the atoms of the other, and (because like charges repel) they push each other away. This allows the positive charges on the massive protons in the nuclei of the two atoms to "see" one another, and when that happens, the atoms push each other away, and thus the objects never actually touch one another. We perceive this as "physical contact" between the two objects, and it is responsible for the fact that when we sit on a chair, we do not fall through it onto the floor, or through the Earth to its center.
Van der Waals forces can also be attractive under certain circumstances; they result in the lattice of atoms that make up a crystal, and in fact they are responsible for the "solidity" of all solid objects, attracting the atoms and molecules to one another and holding them still relative to one another.
Van der Waals forces are an example of "residual forces," so-called because they are not the "full strength" of the force, but a situation where there are some attractions and some repulsions that balance off against one another. Another residual force is the strong nuclear force, of which you may have heard, and in older physics books may have been told was one of the four forces. I will explain; but first, I must explain the color force.
The color force is confined; because of the particular way that it operates, we never see its action directly, as we see the actions of the other three forces. (The weak force is very short-range, but still not confined as the color force is.) This force operates between quarks, which are the constituent particles of the neutrons and protons in the nuclei of atoms. There are six kinds of quarks, but neutrons and protons are made of only two of them. The six kinds are called "flavors," so called by a whimsical physicist, and the whimsy (which originally began with their name, taken from an obscure passage in Ulysses by James Joyce) continued with the names of the flavors, which are up, down, strange, charm, top, and bottom (originally, the last two were truth and beauty, but this was too whimsical at last to be kept). Only the up and down quarks make up almost all of the matter we ever see. Strange quarks give the particles that contain them unusual masses; particles that contain charm quarks live unusually long periods ("charmed" life, so to speak). Top and bottom quarks are very massive and not much seen except in high-energy particle accelerator experiments. The quarks also have colors, and since there are three, they were of course named red, blue, and green. There are also three anti-colors, which are simply anti-red and so forth.
The flavors of the quarks determine their charges and their contributions to the masses of the particles made of them; but their colors are the "handles" by which they are held by the color force. The particles made of quarks are called the hadrons, and the subgroup of the hadrons that the neutron and proton are part of is called the baryons. Each baryon is made up of three quarks (there are other hadrons called mesons made up of only two, but they do not come into this discussion), and each of those quarks is of one of the three colors. There is a set of eight colored gluons that mediate the color force, that both binds the three quarks together, and exchanges the colors among the quarks.
From far away (at least, far on the atomic scale), each neutron or proton looks like a single little particle; but if you get up close, you can see it's actually made of three smaller particles held closely together and exchanging colored gluons back and forth. Unlike the electromagnetic and gravitic forces that we are familiar with, the color force does not become weaker with distance; in fact, it gets stronger. This is because its character is different from gravity and electromagnetism; and another strange thing about it is that it is much more like magnetism than electricity. When you try to pull the quarks apart, as they move apart, they are more strongly pulled toward each other, and eventually, you have to exert so much force that there is enough energy to make more quarks. These quarks promptly appear, and associate into a meson or baryon, and go racing off, carrying the energy that was making the force that was interfering with them away and restoring order.
When neutrons and protons are pushed close together, like they are in the nucleus of an atom, then the quarks are close enough to interact a little bit; and the blue one will push its mate in a nearby neutron or proton away, and snuggle up to the other two quarks to which it is attracted. Once the neutrons and protons have all arranged themselves so that all the quarks are happy, they stay that way, and the force that holds them is far stronger than the electric force that tries to push all the positively charged protons apart. Thus we can see that the strong nuclear force is actually a residual force of the color force between the quarks, made of both attraction and repulsion as the van der Waals forces are among atoms.
The last force is the weak nuclear force. I spoke earlier of the up and down quarks that make up protons and neutrons; and you probably wondered just what the composition of them was. It's two up quarks and a down quark for the proton, and two down quarks and an up quark for the neutron. I also spoke of charm quarks "living longer," and of other quark flavors. So what, you probably thought, happens to those other quarks? Well, the answer is that they decay. All of the quarks decay from heavier ones to lighter ones. Even the down quark, which is slightly heavier than an up quark, decays. Apparently, only the up quark is stable. And for a reason called "confinement of electric charge," which means that we see only unit charges in nature, and because of the fact that quarks have fractional electric charges, we cannot have three up quarks; we must have a down with them. That down cannot decay without violating electric charge confinement, so that configuration is stable, and it is the proton. But is the neutron not stable? Not on its own, it isn't! A neutron by itself decays with a half-life of a few tens of seconds into a proton, giving off an electron and an electron neutrino as it does. What really happens is that one of the down quarks turns into an up quark because of the action of the weak nuclear force. For that is what the weak force does: changes quarks into one another. In most nuclei, the neutrons are stable; they are held by their association with the protons from decaying, but in other nuclei, they are unstable, and this is the reason for radioactivity.
Now, the last thing to know about forces is that they are associated with charges; and the charges are associated with conservation laws. And there is a very important theorem that was invented by a woman named Emmy Nother and called "Nother's theorem" that every conservation law is associated with a symmetry. Without going into too much detail, considering this already overly long post, a symmetry is something that remains unchanged when everything around it changes; and in physics, the something is the laws of physics. So, for instance, there is symmetry over translations; that means that if I move from over here to over there, the laws of physics are still the same. This is responsible for the fact that when you walk around a corner, you don't "turn into" a penguin! Well, gravity is easy: it is associated with the symmetries of physics over translation. And, as it turns out, the charge for gravity is mass; so mass, position, and gravity are all linked together into this web as a single thing. And there are symmetries that correspond to the other forces too; and charges and conservation laws to go with them. To understand this all better, I'd recommend a book I liked very much by Vincent Icke named The Force of Symmetry. It was published in 1995 by the Cambridge University Press.
But there is your answer as to what a force really is: it's a symmetry! :)