The primary coil can be made of most anything conductive, but the general preference is soft copper tubing ( the kind used for hooking up the water supply to the ice maker on the refridgerator). For small to medium sized coils, 1/4" dia. by 50' is a good starting point to have. For larger powered coils, 3/8" tubing is preferable. The general configuration of the coil is a flat spiral ( like the electric stove elements) although there are variations on the theme.
There is also a design which is known as the inverted conical. This type of coil starts in the center of the spiral and is wound outward and upward forming a "funnel" shape. By doing this the magnetic field created by the coil is "shifted" more inward, thus increasing the field effect upon the secondary coil. This is known as increasing the "coupling". Increased coupling is commonly used on lower power coils. In general the lower the power of the transformer, the more coupling is needed to get the coil to function well.
The Helical primary coil is wound similar to a spring. This configuration
allows for the fullest amount of coupling to occur with the secondary coil.
This coil type is used in very low power coils and also on the opposite
side of the spectrum, in Magnifier Coil systems.
With my latest coil, I have used both the flat primary and inverted saucer design together. The first several turns are at 45 degrees and the last 10 turns are in the flat configuration. This design allows better coupling near the secondary and it also keeps the outer portion of the primary low to decrease the chance of a primary strike from the coil. Also by having the outer portion of the primary flat, it allows for easier tuning with the present clip I am using.
The most important aspect of the primary coil is the ability to be able
to "tap" the coil for tuning. The spacing between the tubes must be enough
to allow a clip, clamp, etc to be placed upon the tubing and not be close
enough to the other tube to start arcing to it. The way most primaries
are secured to the base plate is by the means of a supporting structure
that had been notched, drilled, or slotted in such a manner to place the
tubing into the holes or notches. Plexiglass and wood are the most commonly
used material for the support structure. Polyethelene plastics are just
as equally well usable for the supports, and besides it drills alot easier
than plexiglass.

Inverse
Conical Primary on Plexiglass supports
Flat Primary with Secondary coil in place

Flat Primary with Strike ring
Helical Coil on a Magnifier coil system
http://www.geocities.com/CapeCanaveral/Hangar/3108/primary1.html
Primary with the 45 degree inner turns and flat outer turns plus the modified
battery clamp