Statics is the branch of engineering physics that deals with objects at rest. Newton's Second Law of Motion forms the entire mathematical basis for the study: it says, quite simply, F = m * a . Statics takes this in several directions, arriving at a cornucopia of useful equations for discerning the internal forces on a structure at rest.
First, a is reduced to zero, via Newton's First Law of Motion. This means that regardless of the mass of the objects, something is holding them still, and there is no net acceleration. That something must be an opposing force (see Newton's Third Law of Motion), which leads us to the fundamental equation of statics:
ΣF = 0
That's it. "The sum of the forces equals zero." Not only that, but you get a freebie dealing with angular momentum as well. Since the object is neither moving nor spinning, you can assume that there is also no net torque on it; here we represent torque with a capital "M" for "moment". Because ΣM = ΣF * d, and ΣF = 0, we know that
ΣM = 0
"The sum of the torques equals zero." Again, utter simplicity.
These two rules are true for every single direction. For instance, all of the parallel forces in the horizontal direction sum to zero; so do the ones in the vertical. So do the ones canted 33o. All of them sum to zero. A rock sitting on a table has two forces on it, both in the vertical direction: gravity, trying to accelerate its mass downward at 9.8 m/s2, and the table, trying to hold it still, pushing back with the exact same force. If the rock weighs ten kilograms, then the downward force is 98 Newtons, and the table is pushing back with 98 N of force also. Somewhat boring, but that's the way things go in one-dimensional statics.
Because of the work of Descartes and his proofs about perpendicular vectors, we can choose one axis for each dimension we're working with (x; x and y; or x, y, and z -- all mutually perpendicular) and do our summations in 1-, 2-, or 3- space. Our new rules become
ΣFx + ΣFy + ΣFz = 0
ΣMx + ΣMy + ΣMz = 0
This is still read as "the sum of all forces and the sum of all moments are each equal to zero." The rest of statics is just figuring out how to calculate the various forces. Let's look at our rock again. Let's suppose there's a very stiff breeze exerting a force of 3N on the rock, and a prehistoric pseudo-rodent trying to push this rock across the table, perpendicular to the direction of the wind, with a force of 4N. He's not very bright, is he? Well, that's why Allotheria are extinct. Always going around pushing on rocks... So. Let's declare three axes: x will be the direction of the wind, y will be the pseudo-rodent, and z will be gravity. The table pushes back with 98 Newtons and nullifies gravity, and something holds the rock in place against the wind and the prehistoric pseudo-rodent.
It's friction, for those of you impatient with statics and eager to move on to a node about Butterfinger McFlurrys or some such. By the Pythagorean theorem, we know that the force of friction is 5N (draw a free body diagram if you don't believe me). From this, an engineer can apply the static friction equation (Jeeves has an outstanding writeup there) and tell you about the interesting properties of the underside of a rock. What this tells us about engineers is left as an exercise to the reader.
If statics isn't exciting enough for you, perhaps you'd like to learn about dynamics, where things move. It's really not that much more complex until you start doing the math.