Composting requires nitrogen, carbon, water, and oxygen to work properly. The nitrogen, which comes from green (or otherwise colored) vegetation matter, helps the microorganisms grow and reproduce so they can oxidize the carbon. Carbon is used as an energy source for microorganisms and usually comes from brown, dry compost ingredients like crushed up dead leaves. Water keeps things moist, but not so wet as to drown the microbes, and oxygen oxidizes the carbon, which is the heart of the decomposition process.
Certain ratios of these materials in your typical plastic compost bins give the microbes all they need to work fast, which is what causes the compost to heat up. When this happens, water is evaporated, and oxygen can get depleted quickly, so more air and water may have to be added to keep the process going. Adding oxygen is a matter of stirring the compost, or simply turning the crank on rotating compost bins, while water may need to be manually added before stirring or rotating.
The best rate of composting occurs when the carbon and nitrogen are present in ratios of about 30 to 1. Mixing equal volumes of green matter like grass clippings with brown matter like dead leaves gives you something close to the ideal carbon to nitrogen ratio. But even if you don't get the mixture just right, it will still compost - just more slow. If this doesn't bother you, then it isn't a problem.
The composting process also relies on microorganisms to break down organic matter. Some of the most common microbes that are found in active compost include bacteria, actinomycetes (which break down cellulose-heavy things like bark and paper), molds, fungi, and yeast, all of which break down materials that bacteria are unable to. Protozoa are found in compost, as are rotifers, which keep bacteria populations under control. In other words, it's a whole little ecosystem right inside your composter.
You can add to this little ecosystem by introducing earthworms to your compost. They ingest partially composted material, continually aerate the mixture, and create drainage tunnels as they work their way through the compost. And earthworm manure is sort of like super-compost, concentrating organic nutrients even more.
If your compost slows down, there may be many reasons for it. The process almost always slows down during winter, unless measures are taken to speed it up. Under non-winter conditions, however, a lack of sufficient healthy microorganisms is the main reason the composting process slows. Other common reasons for a slowdown in the composting process are a lack of moisture, or too much moisture.
With the modern compost bins for sale today, as long as you steadily add a variety of kitchen scraps, toss in some crumbled dead leaves every week or so, stir or rotate regularly, and keep it from drying out or becoming too wet, chances are, your compost will form right on schedule, giving you nature's best organic fertilizer like magic, right from stuff you would have thrown out.