A Record Group is a way of organizing records in an archive. Records from the same source are stored together and organized in the same way. Records from a different source, even in the same archive, may be organized very differently, and would be in a different Record Group. Record Groups are usually numbered, not named. And the number is specific to the archive. For example, for NARA Record Group 58 is records from the Internal Revenue Service.
In the US, when someone refers to a Record Group, they often are refering to NARA, but it is best to always identify the archive when refering to a Record Group.
This is a good list of NARA Record Groups: http://www.army.mil/cmh-pg/reference/nararg.htm