How did they kill Agent Smith?

There were multiple Smiths that had assimilated on the other entities inside the matrix, both software and human.

The primary difference in the end was that Neo was hardwired to the source and not jacked in from a regular pod or an illegal back door entry.

When Smith assimilated on Neo, the machines get a hold of Smith's replication programming. This enables them to target and destroy the Smith program which causes the replication to be undone across the Matrix.

In short, Smith had turned into a replicating worm which the machines couldn't get a lock on until the replication occurred on an entity (Neo) connected directly from the source.

File creation date. That would tell who was the copy and who was the original.

DELETE Top 1 Agent
from tblAgents
Order by FileCreationDate ASC

The above is pseudocode, but logically it's SQL that would be used to delete the main agent from a table of Agents.

That assumes they don't have a Primary Key field, where they could just delete him by his number. Even when creating a duplicate record, the Primary Key can't be duplicated, so they could just delete the agent that has Smith's Primary Key from the table.

Hmm... my answer is that, much like Johnny Bones' answer, there is a table of Agents. Agents can refer back to themselves, so you could have something like:

ID     Agent        Refers to
1      Smith Prime  NULL
2      Brown        NULL

And so on, until you get to the Smith clones:

ID     Agent        Refers to
1      Smith Prime  NULL
2      Brown        NULL
3      Smith Clone  1
4      Smith Clone  1

And so on. Now, we can assume that, by deleting "Smith Prime", the other Smiths will become what are known as "orphaned records". In a badly-designed database system these would still be allowed, they just wouldn't work properly. We have to believe that the Matrix is actually well-designed and deleting "Smith Prime" would have a cascading effect, or possibly a trigger, that would delete the clones too.

However, this doesn't answer HOW we are able to delete "Smith Prime". In sql pseudocode, as Johnny Bones pointed out, the Machine God Giant Floating Baby Head Thing could simply have found and deleted his record. We have to believe that Smith Prime is stopping this somehow.

When people are physically jacked in to the Matrix through the Matrix's own systems, their bodies are able to be taken over by the Matrix - or more rather its Agents. We've seen this a few times in the movies - random person becomes Agent. This doesn't seem to be able to happen to the Zion operatives as they're wired in a different way. But when Neo jacks in via the Machine God Giant Floating Baby Head Thing, he's wired in directly through the Matrix's own systems. Even better (or worse, depending on your viewpoint), he has the full attention of the Matrix - both inside the Matrix and in the physical world.

When Smith Prime copies himself onto Neo, he also copies himself onto a mind directly connected to the Matrix - one which the Machine God Giant Floating Baby Head Thing has influence over. Smith Prime is vulnerable in a more frail, human mind, at least temporarily. Neo's body, with Smith Prime inside, can be disconnected and killed in short order.

In short, Neo was bait and Smith was arrogant. Had it been a clone, then it probably wouldn't have worked.