Short: "s" means set.
Long:
As applied to a file, chmod g+s is used for setgid. That is, if the file is executable and has the group-s bit set, it will run with its group set to match the group-ownership of the file. The feature has been in Unix for a while (see for example the 6th Edition manual page).
Directories are harder to document, because the behavior may depend upon the system.
POSIX does not specify a behavior, saying only
Conforming applications should never assume that they know how the set-user-ID and set-group-ID bits on directories are interpreted.
For AIX and HPUX, nothing is mentioned for directories
For Solaris
For directories, files are created with BSD semantics for propagation of the group ID. With this option, files and subdirectories created in the directory inherit the group ID of the directory, rather than of the current process. It may be cleared only by using symbolic mode.
although (noting "BSD semantics"), this behavior was not described in the SunOS 4 manual page.
In modern systems, FreeBSD, OSX and Linux provide the "BSD semantics", but NetBSD and OpenBSD do not mention it.
Further reading (interestingly, no accepted answers):
