Process context is the identity and permission set a program uses while it runs. On Unix-like systems this usually means the user and group that started the process, plus any elevated rights, file access, or network capabilities attached to it. In practice, every command, file read, and connection an application makes is filtered through that context.
It matters in cyber security because the same vulnerability can have very different impact depending on which process context is involved. If an attacker achieves command execution in a low-privilege service account, damage may be limited; if the process runs with broad access, the attacker can read sensitive files, modify configuration, or pivot further into the system. Defenders use process context to judge risk, harden services with least privilege, and investigate logs for suspicious activity originating from a specific application account. In attacks against mail servers and other infrastructure, seeing commands run in the application's own process context is often the first sign that exploitation succeeded.