An exploit is a general term for any method used by hackers to gain unauthorized access to computers, the act itself of a hacking attack, or a hole in a system's security that opens a system to an attack.
The term is very flexible and can be used both as a noun as a verb. As a noun, the exploit is the hole in the system that the hacker used to make the attack. Quite often, this is an OS vunerability from an unpatched server. As a verb, it refers to the act. For example, you might hear "the hacker posted details of his exploits on his blog to show just how easy it was to break into XYZ's servers."
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