The 1998 & 1999 Close AM cent is so named as the A and M in America on the reverse of the coin nearly touch at the base in this variety. The regular or usual 1998 & 1999 proof cent has a space between those letters as you can see in the photos.
The business strike coin of this same year has the letters nearly touching and how the Close AM Cent came about is that one or more dies that were meant for business strike coins were mistakenly used to make proof coins.
The 1999 Close AM was discovered first and there appears to be quite a few of them although the total numbers are not known. On the 1998, there seems to be far less of those and they maintain a higher price.
There is another similar variety that can be found in the 1992 D business strike cent which is quite rare. One sold that was certified by Professional Coin Grading service as a PCGS MS64 for $20,700 at the Heritage Auction sale at the Florida United Numismatists summer convention.
There are also a couple known Wide Am cents in business strike coins made from reverse proof dies in 1998, 1999 and 2000. These tend to be fairly common and can often be purchased for a few dollars.