Primary Capacitor
The primary capacitor stores the energy from the powersupply to enable it to be released in a single burst (bang) when the spark gap fires.
As in most coils the primary circuit is running with voltages of 1kV - 22kV, the capacitances need to have dielectrics (insulation between their plates) capable of withstanding this sort of voltage. Usuall values of primary capacitor are 6nF - 10uF depending on coil size and primary voltage.
For maximum efficiency the capacitor needs to deliver all of its charge in a very short period of time, this is mostly restricted by its effictive series resistance(ESR) pulse currents of 400A are not unknown in Tesla coil primaries.

High Voltage Capacitor 

Very Expensive, Very good
performance, difficualt to change value, 
high capacitance per unit volume, Good ESR.

 MMC (Multi Manufactured Capacitor)

A high voltage capacitor made from many smaller off the shelf capacitors.
Cheap(ish) to produce, Requires spreader resistors, easy to change values,
can be made virtually any size. Reasonable ESR if constructed well.

DIY Capacitors

Difficualt & time consuming to make, Physically large for a given capacitance
Good ESR if constructed well, can be very cheap. e.g. beer bottle caps.

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