So why is it the if you come to complete stop your rpm will remain between 500 and 1500.
Here, I am assuming you are talking about an automatic transmission (AT). The major part of the AT which allows this to happen is the torque converter (TC). The heading of Wikipedia article sums this up very nicely:
In modern usage, a torque converter is generally a type of fluid coupling (but also being able to multiply torque) that is used to transfer rotating power from a prime mover, such as an internal combustion engine or electric motor, to a rotating driven load. The torque converter normally takes the place of a mechanical clutch in a vehicle with an automatic transmission, allowing the load to be separated from the power source. It is usually located between the engine's flexplate and the transmission.
The key characteristic of a torque converter is its ability to multiply torque when there is a substantial difference between input and output rotational speed, thus providing the equivalent of a reduction gear. Some of these devices are also equipped with a temporary locking mechanism which rigidly binds the engine to the transmission when their speeds are nearly equal, to avoid slippage and a resulting loss of efficiency.
This answer I wrote describes how a TC works, so may be of value here.
And how does acceleration occur?
Since you used the engine tag, I'll start at the front of the vehicle.
Engine - The vehicle needs a power source to move the vehicle and to power the different accessories (alternator, power steering pump, brakes, etc) throughout the vehicle. When designing a vehicle, engineers design an engine which not only fits in the space provided, but also design it so it will produce torque/horsepower specific to its needs.
Transmission - The transmission allows the vehicle to utilize the torque/horsepower of an engine to full advantage through torque advantage and through reducing engine speed at higher engine speeds. (NOTE: I'm including the torque converter/clutch as part of the transmission.)
Differential - Allows for a change in direction of the rotation produced from the engine (through the transmission) as well as a means of gear reduction/torque multiplication.
Tires - This is where the rubber meets the road ... literally. Tires also create a torque reduction with regards to their overall diameter. The taller the tire, the more torque it takes to turn the wheel/tire assembly. Again pulling from a Wikipedia article:
Torque, moment, or moment of force (see the terminology below) is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist to an object ...
Loosely speaking, torque is a measure of the turning force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque (turning force) that loosens or tightens the nut or bolt.
- Tires (cont.) - If looking at tires from the side, you were to draw a line between the center of the tire to the ground (radius of the tire), this then becomes your rod length. As this rod becomes longer (increase tire radius), it would become easier to turn the tire by moving it directly (rolling the tire directly by hand). Conversely, if you are twisting the rod from the center point, as the drive axles do, it becomes harder to turn the axle as the rod becomes longer. The differential's torque multiplication helps overcome the tire's torque reduction.
The engine produces the torque needed to motivate the vehicle. When sitting at a stop, the engine is happily purring along at idle. It can do this because either the TC or the clutch has allowed it to be decoupled from the rest of the drive train. As you prepare to pull away from the stop, you press on the gas pedal and the engine RPMs increase. If you have an automatic transmission, the torque converter allows slippage between the engine and transmission while still transmitting torque and multiplying it at the same time. The TC will allow the slippage until it reaches its stall speed. An excerpt from the Banks Power website:
Even under light loads, a vehicle with an automatic transmission will start moving as soon as you take your foot off the brake. The stall speed comes into play under all load conditions. When we talk about stall speed, we're referring to engine RPM. If the vehicle isn't moving by the time the impeller reaches the stall speed, either it will start to move, or the engine RPM will no longer increase. In other words, stall speed is the engine RPM at which the torque converter transfers the power of the engine to the transmission.
A vehicle's engine is designed with a certain torque/power band so that, along with the transmission, the engine will be allowed to operate within that torque band to allow the vehicle to move. If you try to apply too much load (taking off from a light with a standard transmission, for instance) without enough engine RPMs, you'll stall the engine (NOTE: Engine stall has nothing to do with TC stall) and the vehicle won't move. Slipping the clutch in this situation is analogous to how the torque converter allows a slippage. By doing so, the engine is allowed to get into its designed toque/power band and the vehicle can then start moving. Once the vehicle is moving and the engine operates above the TC's stall speed, the engine is then motivating the input of the transmission at the same general speed as it is running (there is a bit of slippage there until TC lockup ... but that's another story entirely).