In this particular example, the C20 is the index or reference number used in the encyclopedia of chess openings to identify the king's pawn opening. The ECO is a huge database of chess openings and the alpha-numeric codes are helpful for indexing them.
The Encyclopaedia of Chess Openings (ECO) was originally a 5 volume book on chess openings. Volume A covered Flank openings, B semi open games, C open, D closed and E indian openings. Each volume was then sub divided into 100 different categories from 00-99.
Originally posted by mrbluetone Excellent...I didn't know there was such a reference index, let alone that I was playing them by accident!
That's no accident. There is basically no way (except resigning before move 1) to not play an opening from ECO. Even if you do something daft like 1. f3, that's covered under A00. And if you play anything even remotely reasonable, it will have been played before and fall under one opening or another - possibly a rare one, but a known opening with a non-A00 ECO code nevertheless.
A nice feature on this site is being able to look up all your finished games sorted by their ECO codes, using My Games-Search For Games-Enhanced Search. So if you want to see how well you have done with a particlular opening you can pull out all games that you have played by putting in the ECO code for that opening.