An oil rig can come in many variations depending upon the size and scope of the drilling operation. At one end of the spectrum you have the extremely high-tech deep water drilling ships designed to drill as deep as 35,000 ft. At the other, there are small land rigs which aren't able to drill more than 2,000-3,000 ft. All are commonly recognizable due to the derrick, a towering framework of steel beams.
No matter the size, all rigs have three primary purposes: Weight, Rotation, and Circulation.
As far as weight is concerned, the rig doesn't 'push' at all. Gravity acting on the mass of the drill string itself provides adequate weight to the bit. The drill string is supported by steel cable that is wound around a drawworks. The rig uses the drawworks to control the amount of weight that is put on the formation. The drawworks acts as a large brake during drilling, and as a way to pull the string back out of the ground.
There are two ways rotation is applied to the drill string. The older method is by using a kelly. A kelly is a special length of drill pipe which instead of being round is usually octagonal. This octagonal pipe fits into a corresponding hole in the rotary table. The rotary table can be rotated, thus imparting rotation to the drill string. The other, more modern, and more efficient, way to rotate the string is by using a top drive. A top drive screws directly into the top of the drill pipe and both supports the string, and provides rotation.
A rig must have a way to remove the cuttings created by the drilling bit, and to cool the bit. This is accomplished by using between one and four large pumps. These pumps pump fluid down the center of the drill pipe. The fluid returns to the surface by means of the borehole annulus. A complex system of pits is necessary to control and contain drilling fluids both before, and after the fluids are pumped downhole.