Coils are used to heat up the Heated Furnace. A Coil can be placed on all sides of a heated furnace and the heat provided by the Coils can stack. So the more Coils next to a heated furnace, the faster it will run. Different tier Coils can be used together at the same furnace or be used by multiple furnaces at the same time (except the Kinetic Coil, which can only be used by 1 furnace due to the way the BC laser API works).
The Basic Coil burns solid fuels like a regular furnace, each Coil provides 1/5th the speed of a furnace but fuel lasts 6x longer in a Coil.
The Lava Coil uses provides more speed than the basic Coil and burns lava instead of solid fuel, it has an internal buffer of 3000mB, right-click on it to see how much lava it has stored.
The Kinetic Coil is by far the fastest Coil, it is powered by nearby BuildCraft lasers, the more lasers the faster the connected furnace will be, 1 Coil per furnace is more than enough, it can accept as much power as the lasers can provide it (furnace speed is capped at one item every tick though).
When using a Kinetic Coil an auto-eject upgrade for the Heated Furnace is highly recommended as it produces items insanely fast.