The easiest way to answer this is to say that the most recent American involvement in a war in Iraq started in March of 2003. The US involvement did not officially end until President Obama declared the war to be over in December of 2011.
However, a case can be made that the war in Iraq started much earlier than this. You can argue that the war started in 1990 or that it started with the 9/11 attacks. The 9/11 attacks made the George W. Bush administration worry about further attacks and they came to feel that Iraq might be a sponsor of such attacks. Some scholars feel that President Bush was preparing for the war in Iraq from very soon after 9/11. This is connected in part to the 1990-1991 Persian Gulf War. At that time, the US-led coalition drove Iraq out of Kuwait but did not overthrow Saddam Hussein. From that time on, there was bad blood between the US and Iraq and worries about Hussein's WMD programs. These worries combined with the worries caused by 9/11 to push the Bush administration to invade.
So, the actual invasion of Iraq began in 2003, but you can argue that the prelude to the war began long before that.