Add water
Micro-organisms that convert the materials to compost need water. The moisture content of the material should be 45%. It will feel damp but not wet.
Things like too much and too little water will slow down your compost too. If it's the raining, cover the piles with plastic so it doesn't get too wet. In the dry heat of summer, mist it down occasionally to keep it from drying out. How do you know if it's too wet or not wet enough? Keep it to where a handful of compost material when squeezed will hold its shape, yet not wring out any water.
Add oxygen
The material does not have to be layered. It should be turned every two weeks or so to re-aerate it.
Lots of air = lots of organisms = rapid decay
Add micro-organisms
These are to be added along with the raw materials. They can be provided by mixing a spade full of garden soil or some old compost through the materials.
Preparation
Turn it every two weeks or so and add water if necessary. It should be "cooked" in 10 or 12 weeks.
"Cooked"
The process of composting produces heat. thus you'll hear people talking about things like core temperatures and cold piles. The temperature of an average compost pile in it's core at peak heating runs between 145 to 165 degrees F. The idea is to maintain that heat until it will not heat any more, cooked compost runs about 100 degrees F.
Mix it into the soil or spread it on top.