In the Matrix, Mr. Smith is an agent at the beginning of the story. Agents are rule-based programs but why was Mr. Smith removed as an agent? What has he become?
He was disconnected/unplugged as an Agent because of his contact with Neo (this meant that he was no longer an Agent and became his own entity), remember that Neo jumped into his body at the end of the first film. It's stated here in the wiki for Agent Smith.
Shortly after Neo escapes the fight, Smith guns him down. Neo revives, realizes his power as the One, and enters Smith, to destroy him from within.
Then in the second movie.
The Matrix Reloaded
As a result of his contact with Neo from the first film, Smith is "unplugged" in the second film, no longer an Agent of the system but a "free man".
Also he wanted to escape The Matrix as portrayed in his conversation with Morpheus in the first movie.
Can you hear me, Morpheus? I'm going to be honest with you. I hate this place, this zoo, this prison, this reality, whatever you want to call it. I can't stand it any longer. It's the smell, if there is such a thing. I feel saturated by it. I can taste your stink. And every time I do I feel I have somehow been infected by it. It's repulsive, isn't it? I must get out of here. I must get free and in this mind is the key, my key. Once Zion is destroyed there is no need for me to be here, don't you understand? I need the codes. I have to get inside Zion, and you have to tell me how. You're going to tell me or you're going to die.
As an Agent he is already beginning to go rogue at this point.
After being unplugged he became a rogue program with the normal powers of Agents but also the ability to copy himself into other programs and humans instead of being able to hijack them temporarily. Essentially Agent Smith became like a virus as stated here in the wiki.
He still possesses the abilities of an Agent, but instead of being able to jump from one human to another, he is able to copy himself over any human or program in the Matrix through direct contact; this includes humans wired into the Matrix, non-Agent programs with human forms, redpills, and other Agents. Smith retains the memories and abilities, if any, of the one over which he copies himself. This ability is much like how a virus replicates, creating an ironic contrast with the first film, where Smith likens humanity to a virus.
Towards the last film it's explained by the oracle that Smith became essentially Neo's evil counterpart, the Matrix's solution to him. So essentially Agent Smith turns out to just be an evil version of Neo.
The Oracle explains to Neo that he and Smith have become equal in power and that Smith is Neo's negative, a result of The Matrix equation trying to balance itself.
Reloaded spends a fair amount of time talking about programs that don't do what they're supposed to do. Interestingly these programs are regarded by Agents to be as much a threat to the stability of the Matrix as the potential redpills - look at the way their attention shifts during the highway scene, or during any point where an Agent confronts any entity which is clearly subverting the intended rules of the Matrix. The ability for any mind to choose its level of involvement in the Matrix's version of reality causes fluctuations which The Architect is trying to root out.
As a rogue entity, Smith cannot be an Agent. Agents exist to maintain the status quo, and Smith is not interested in maintaining the status quo. Therefore he cannot serve as an Agent in any capacity.