How did the timeline work?

Film.com has an infographic by Rick Slusher:

[click link for a larger image]

Timeline A:

• In Timeline A, the looper hitman Joe successfully “closes his loop” in 2044, executing his time-traveling older self at the prescribed moment, earning his golden payday and proceeding to live out the rest of his life.

Regarding the other key characters in Timeline A, the infographic illustrates a problematic time paradox in the film:

• Despite Old Joe’s execution in this initial timeline, young Cid must still somehow come to lose his mother Sara and, devoid of her positive influence, develop into the angry, telekinetic, all-powerful gangster kingpin known as the Rainmaker in order to send Old Joe back in time to close his loop in the first place.

Timeline B:

The moment of Joe’s successful loop-closing in Timeline A instantly launches Timeline B:

• Joe’s 30-year post-looper retirement inevitably leads to his eventual wife’s tragic murder in 2074. The trauma inspires Joe’s older self to change his life’s trajectory by escaping his own self-execution in the past to hunt down the child who will become the Rainmaker, whom he holds responsible for his wife’s murder.

Timeline C:

Therefore, Joe’s altered moment of execution in 2044 – this time a narrow escape rather than a successful kill – instantly launches Timeline C:

• Old Joe seeks to save his wife by murdering the Rainmaker as a child, whereas Young Joe understands this attempted murder will paradoxically bring about the very outcome Old Joe is striving to avoid.
  
  Young Joe’s final decision to commit suicide alters everyone’s destiny: As both he and his older self cease to be, Cid is allowed to grow up under his mother Sara’s positive influence and never develop into the monster he might have been.

Based on your question, we're ignoring the fun times around Seth at the start of the film, and focusing on Joe. Having seen the film twice, there are some nuances I missed the first time through, so I'm altering my answer. There are three timelines that matter:

• First, there's the original timeline where Old Joe is taken, his wife is killed, but then he fails to overpower the mob henchmen. He is sent back slightly late, and is promptly killed by Young Joe.
• Second, the timeline is similar to the above, except Old Joe overpowers the henchmen, and decides to go back to kill the Rainmaker. This is where the bulk of the movie happens. As the ramifications of Old Joe and his mission ripple through the present, they impact the future history of Old Joe. Young Joe carves 'Beatrix' into his arm to arrange the meet with Old Joe. Old Joe kills the first child, then remembers a future where he and his wife had a kid. It seems that once Young Joe learned of this, it impacted his original-timeline decision to not have children with his future wife.
• Third, Young Joe kills himself, Old Joe ceases to exist. Sarah is safe, and raises the widdle Rainmaker to not be a monster.

The director acknowledges there are some oddities around the timelines, as the Rainmaker is described as having a fake jaw and seeing his mother killed in front of him, which would suggest that we always see timelines where Old Joe shot widdle Rainmaker.

Within the film narrative, I think it's stating that one of the ramifications of timeline 1 above was that because Joe knew his loop would be closed, that he then was able to successfully overpower the henchmen and choose to go back in time. Thus, he created the 2nd timeline, because the causality loop of him killing himself wasn't tight enough, due to the possibility of overpowering the henchmen.