Can you tell me exactly what “proof” condition means in a gold coin’s description?

“Proof” is a designation given to coins specially-made for the collector's market which have exceptional sharpness of detail and, in mint state, typically feature a mirror-like finish. These coins are not intended for circulation.

Instead of an actual grade, "proof" designates the manufacturing process. However, the term does imply Mint State condition unless otherwise noted in it's grade.  A coin may be a "proof" strike, but still be in less than Mint State condition.  How can this be?  Because proofs that are mishandled by collectors or entered circulation at some point.  As an example, you might find a "PR-58" coin on the market.  This is a coin minted using the special "proof" process, but it's condition is now below the minimum Mint State grade of 60.  How does this happen?  One possibility is that the coin was needed in hard times which caused it to enter circulation, thereby exposing it to normal wear seen on non-proof coins).

Conversley, there are some coins that although they were intended for general circulation were put aside early on because they exhibited the same deep, mirror-like finish of a proof strike even though they are not genuine “proofs”.  This is called "proof-like" and while these coins can command higher prices than their lower-grade counterparts, they will not command the same value as a genuine proof strike, all other things being equal.