There are multiple variants of exploits, a common term is 'remote exploit', which refers to an exploit that can take advantage of a security vulnerability remotely, over a network. A 'local exploit' on the other hand can only increase privileges on a system where some kind of local access is already permitted.
Normally a single exploit can only take advantage of a specific software vulnerability. Often, as such an exploit is published, the vulnerability is fixed and the exploit becomes obsolete for newer versions of the software. This is the reason why some blackhat hackers do not publish their exploits but keep them private to themselves or other malicious hackers. Such exploits are referred to as 'zeroday exploits' and to obtain access to such exploits is the primary desire of unskilled malicious attackers, so called script kiddies.