There are three common types:
Ceramic - Made with various ceramic dielectrics which generally have a trade-off between capacitance per unit size and temperature stability.
Film - These have film dielectrics such as various plastics (polycarbonate, polypropylene, polystyrene, polyester, etc.) and paper (generally obsolete except for oil-filled paper).
Electrolytic - Polarized types. Largest capacitance values. Aluminum is cheapest and most common. Tantalum is more expensive but has better performance.