A vulnerability exploitable with network access means the vulnerable software is bound to the network stack and the attacker does not require local network access or local access. Such a vulnerability is often termed "remotely exploitable". An example of a network attack is an RPC buffer overflow.
Quote from the CVSS specification
Vulnerabilities with this tag were given a CVSS rating as part of the requirement to be included into the National Vulnerability Database. You can learn more about what the individual scores mean in the CVSS specification document.