To answer in a more simple fashion: the electrons in the wires feel repulsion from the other electrons. When no current in moving around, they are in a state called equilibrium. Essentially, the electrons in front of and behind our electron - let's call him the "test electron" - are stationary, so he's roughly stationary too. The forces from his neighbors all balance out. Technically, thermal energy and random fluctuations around him make him jiggle around a bit, but overall he and his friends and just sort of milling around in the same area. This is the case when the circuit is open and no current can flow.
Now, we close the circuit, completing it, and perhaps attach a battery or apply a current of some sort. In the instant this is done - mere fractions of fractions of seconds - nothing happens to test electron (assuming test electron is in the middle of the circuit). The electrons around him are also basically still for this tiny, tiny fraction of a second (unnoticable to our human senses). However, a change quickly moves through the circuit. Electrons near the battery are tugged or pushed, and their electric charge then pushes and tugs their neighboring electrons, and very quickly this force is propagated and felt through the entire circuit. Note that the speed the force travels is very quick, faster than the speed of the actual electrons, but not infinite. Electromagnetic forces are propagated by light waves, which move extremely fast.
As noted above, how much current you get is determined by Ohm's law:
V = IR or I = V/R
Here, I is our current, V our Voltage (presumably from the battery) and R the resistance of the circuit.
To think of it another way, it's like pushing a car. Technically, when you lean over and push the back bumper there's a chain reaction moving from the back of the car toward the front and the whole thing lurches forward bit by bit. However, the electric forces between atoms and molecules in the car transmit very quickly (basically close to the speed of light), so we basically see the whole car move at once.
Some people, however, take this too far and try to image building a giant rod of metal from here to another star system. They think that since they can jiggle a human sized rod and the whole thing appears to move instantaneously, that we could do the same thing to communicate between here and distant stars faster than the speed of light. However, at interstellar distances, the force that moves the atoms has to go so far (light years) that the jiggle would actually be noticeably delayed from getting from the front of the rod to the end. The force can only travel - at maximum - at the speed of light, so at several light years of distance is would take several years for the force to jiggle the end of the rod. Crazy, huh?
Hope that makes sense!