"Metal clay" is a generic term for a material made of microscopic particles of precious metal held together with organic binders and water to create a clay-like substance. When the metal clay is fired at high temperatures, the metal particles sinter into solid metal, and the organic binders and water burn off, leaving just the pure precious metal.
PMC is one of the two major brands of metal clay products. The other major brand is Art Clay. Both are manufactured by Japanese companies: PMC ("Precious Metal Clay") brand products are made by Mitsubishi Materials Corporation of Japan, and Art Clay brand products are made by Aida Chemical Industries, Ltd.
The product lines are quite similar for the most part, although each brand includes some products not in the other brand's line. For example, PMC has an original or "standard" formula of metal clay that must be fired at 1650F for two hours in order for the metal particles to sinter fully, and it shrinks about 30% (vs. approximately 9% to 15% shrinkage for medium- and low-fire metal clay formulas). Art Clay has a Silver Overlay product that can be used to apply pure silver accents to glass, porcelain, etc., an Oil Paste product designed specifically to join together already-fired metal clay pieces (which are fine silver), and slow-dry metal clay formulas that provide longer working times (for example, if you want to braid "snakes" without having them crack from drying prematurely, or if you are working in a very dry climate).
Different metal clay artists tend to prefer one brand over another, but it's mostly a matter of personal taste (the Coke vs. Pepsi analogy is very apt), and often the preference is based on which brand you used first when you were learning to work with metal clay. There are some minor differences in shrinkage and consistency. Many of us use products from both brands for different purposes.
The most important thing to remember is that, after firing, finished items made from either brand are identical in precious metal composition: pure silver (AKA fine silver, .999 silver, or 99.9% silver) or 22K gold.
If you are interested in more information about the similarities and differences of PMC and Art Clay products and the typical application for which a particular product is well-suited, feel free to take a look at my Squidoo lens called "Metal Clay Brands and Formulas: Precious Metal Clay (PMC), Art Clay Silver and Gold, and more!".
Hope this helps!
Margaret Schindel