Is Agent Smith a bug in the Matrix or a virus?

Rather than going by technical, real-world definitions, let's use the in-universe description of a virus:

Agent Smith: I'd like to share a revelation that I've had during my time here. It came to me when I tried to classify your species and I realized that you're not actually mammals. Every mammal on this planet instinctively develops a natural equilibrium with the surrounding environment but you humans do not. You move to an area and you multiply and multiply until every natural resource is consumed and the only way you can survive is to spread to another area. There is another organism on this planet that follows the same pattern. Do you know what it is? A virus

By this description/definition, Smith certainly becomes a virus in the later films, reproducing itself and expanding by taking over new areas until it's destroyed everything else.

At the same time, of course, Smith's aberrant behaviour was not intended; it was a fault. Thus it's fair to say that a bug caused Smith to mutate into a virus.

EDITED TO ADD:

Comments have pointed out that Smith's speech was about biological viruses, and the comparison to humans, rather than a definition of a computer virus. However, the point is that, biological or synthetic, he's displaying the same behaviour that he had earlier condemned. He's become the virus he was describing earlier.

While Smith was undoubtedly talking about biological organisms, the dialogue actually doesn't say that; it just says "organism". The same word is currently used in computer science to refer to simulated life, allowing the same definition to apply to digital life.

Given that he fits the definition of a computer virus given in the question AS WELL as the definition of a virus that he had given in the first film, it makes his fate a case of dramatic irony and character hypocrisy - and definitely, he's a virus.

I would argue that he was a runaway process. Up until his end, he was keeping order in The Matrix, just as he had been told. The problem is that he had somehow transcended his limitations, and was willing to use all of the resources, including those sustaining his fellow residents of The Matrix, to do so.

A ``runaway process'' is a process that enters an infinite loop and spawns new processes. This can cause an overflow in the proc table that causes other processes to fail with the No more processes: error message.

I know this is not a direct answer to your question, but there's actually a pretty convincing theory that agent Smith is THE ONE! You should definitely check out The Film Theorist's video on this topic.

Considering Smith calls the Oracle "mum" (it is also said that the Oracle "created the One"), it would make sense to assume that he's a program written by the Oracle, exploiting some bug in the Matrix, which allowed him to disobey the rules ordinary agents have to follow. Bug-exploiting programs are usually called viruses, so I guess he could be called a virus then.

But hey, it's just a theory ;-)

At the end of the first Matrix, Neo dives into the body of Agent Smith, becomes him, and explodes him. This is the last time that Agent Smith behaved like a normal agent before he began behaving virulently in Matrix Reloaded. You can be certain that at this point in time, Neo overwrote Agent Smith's code. To support this claim, in Matrix Revolutions, during a final confrontation, Agent Smith rants about how something in Neo may have been written onto Agent Smith, and speculates that this may have been a cause for his transformation.

In the first Matrix, Agent Smith functions like an anti-virus program, and Neo functions like a hacker / virus. By Matrix Revolutions, Agent Smith is behaving like the virus. So I would say Neo deliberately reprogrammed Agent Smith to be a virus, and that he is therefore definitely a virus, designed by Neo, and not a bug that happened on its own. By Matrix Revolutions, Neo is rescuing the matrix from the Smith virus, and therefore himself behaving like an anti-virus program. That means Neo underwent a transformation, and Agent Smith underwent a transformation. Character transformations are deliberate and common in well-written cinema, and this turning of the tables becomes more meaningful if they precisely exchanged roles of virus and anti-virus (another reason to conclude virus, not bug).

Lastly, I'd like to point out that Neo's plan to rescue the matrix from Agent Smith and strike a pact of peace with the matrix was a decision he made much earlier. At the beginning of Matrix Reloaded, Neo has an extensive conversation with The Oracle in which she insists that he has already made a decision, but isn't ready to understand his decision yet. Although this can be interpreted to apply to multiple decisions (i.e. his decision to save Trinity. Also note that predictions from oracles in Greek mythology could lend themselves to multiple interpretations and sometimes misinterpretations drive the plot of those myths. Also note that Neo's decision to save Trinity correlated with his belief at the time that the prophecy was a lie), it seems fitting that it likely applies to his decision to reprogram Agent Smith to be a destructive virus. At that point in time, he probably didn't have a full understanding of what benefit it would yield to reprogram Agent Smith; since, in the short-term, it seemed that Agent Smith was simply being a greater nuisance than before. It turns out that there was a not-yet-apparent, logical end-game to reprogramming Agent Smith.

Actually I think he is a pretty aggresive antivirus trying to protect the Matrix from the outer threats, the people. :-D

In the first episode of the trilogy, Agent Smith is neither bug nor virus. Smith is a security agent that works in tandem with other security agents in the Matrix software-hardware-wetware system, placed there by design, possibly The Architect or by The Source.

Self-interest was infused into Smith by Neo during his attack on Smith just after Neo's virtual resurrection in the first episode. It was Neo's attack on the system security of the Machine World on behalf of humanity. (Neither Smith nor Neo admit to knowing the mechanism through which self-interest was infused.)

After the attack and through to the end of the third episode, the now narcissistic Smith began to impose an increasing threat to The Source (for certain) and possibly also The Architect. Even at this point, Smith remained neither virus nor bug. Smith had become something for which there is not yet an equivalent in the IT world.

The unexpected weapon at the climax of the trilogy was choice. Even self-interest and an emerging sense of purpose did not make Smith autonomous.

The screenwriters' final statement is that the protagonist's autonomy inevitably triumphs over the antagonist's destructive impulse. A potential deeper meaning may be that Clausius's thermodynamic assertion that the change in entropy of any system is never negative has one metaphysical exception: Human autonomy.

This is an intensely humanistic philosophy, which may have lent to the popularity of the trilogy, and it is a philosophy that has not be disproved outside of the fictional world.

I'd label him neither bug nor virus, just part of the programming. I don't think it's true that his behavior at the end was "unexpected". It's heavily implied (if not outright stated) that the Oracle created Smith and likely knew his path. At one point he refers to her as "mom", and she anticipates and expects the things he does in the final acts.

I guess it's a matter of perspective, but it seems like the Oracle wanted everything that happened to happen to break the loop and try new solutions to the problems of man and machine. Like "the One", Smith was a purposefully crafted response to the problems in the "equation".

He's neither. He's the balance in the equation that The Matrix is fundamentally built on. The Oracle said this in I think Revolutions.

If you are thinking in computer terms I like to think of it like this. The Matrix has basic rules in the firmware that everything abides by. This includes a balance in an equation. The Architect and the Oracle are another example of this equation balance, one is pure logic and about preserving the Matrix, the other is about "feelings" (can't think of a better word) and breaking it down.

When Neo was created on his "rebirth" in the first film the equation needed to be balanced, I'll go one further than I think was intended and say the Matrix reused the last object that was destroyed to balance the equation… Agent Smith.

What happens beyond this firmware isn't controlled as much, this is where Neo has his powers and Agent Smith can copy himself.

Smith is, for lack of a better term, an intentional bug.

The Matrix has two basic tenets

1. It must accept the input of the users
2. Every action has an equal and opposite reaction

In other words, The Matrix is designed to produce the expected outcome of its users. The reason people don't go flying around or bashing walls, is that they literally don't expect them to work that way. Or, to quote the Hindu-ish kid

There is no spoon

The Matrix can't actually enforce gravity. Or stop you from punching through a wall with your bare hand. That's why people like Morpheus can make long jumps and Neo can fly. Literally, if you expect it to happen, it can happen (likely only applies to places where your expectations can shape the outcome, or Neo wouldn't have needed the Keymaster).

The problem for The Matrix is that, the larger the deviation from reality, the larger the breakdown that occurs as it tries to balance things. Part of the deal The One must make to save Zion is that they break down whatever The One is making go haywire and neutralize it, and then reboot The Matrix. Because of the size and scope of The Matrix, there's no way to figure out what the next One will break, or how The Matrix will respond.

The One is, essentially, both the cause and solution of the bug. Zion is, simply put, a way to keep motivating The One to fix the bug and keep The Matrix working.

The catch is, with the second movie, Smith begins to behave like a virus. But a better explanation is that Smith is not a virus per se (where his goal is to keep replicating) but the bug corrupting everything around it. A better term might be something like a buffer overflow, where a bug in a program allows it to exceed its limitations and corrupt other programs around it.

ORACLE: Everything that has a beginning has an end. I see the end coming. I see the darkness spreading. I see death. And you are all that stands in his way.

NEO: Smith.

ORACLE: Very soon he's going to have the power to destroy this world, but I believe he won't stop there; he can't. He won't stop until there's nothing left at all.

NEO: What is he?

ORACLE: He is you. Your opposite, your negative, the result of the equation trying to balance itself out.