Colin Matthews, formerly Chief Executive of Severn Trent plc and Director of Technical Operations at BA, will take over the role.
Nelson has been chief executive only since 2006, arriving from grocers Sainsbury's, where he was marketing director, in 2005.
Stephen Nelson said: "With the opening of Terminal Five, and the completion of the CAA Price Review for the next five years, BAA really is at the start of a new beginning.
"Whilst BAA faced considerable challenges during my period as Chief Executive, I am immensely proud both of the way the company responded to those challenges, and refocused its efforts to put the passenger first.
"The opening of Terminal Five is a landmark occasion, both for BAA and for the history of aviation in this country. To have delivered this majestic building on time, and on budget, reflects all that is best about BAA and those who work for it, and I wish Colin Matthews and BAA all the best in driving forward that new beginning.”
Colin Matthews, who led the reorganisation of Severn Trent Water said, "Terminal Five shows the future BAA wants for all its passengers and I am determined to continue to raise standards across all our airports.
"By committing to invest £10bn across our seven airports in the next ten years, our shareholders have already shown that they too share that vision.
"No-one under-estimates the size of that challenge, or the difficulties we face on a daily basis in delivering it, but, equally, I am fully aware of how significant transforming our airports is for the future of this country and I am immensely proud to be joining BAA and its leadership at such an important time in its history."
BAA's parent company, Ferrovial, yesterday hinted that the sale of one of its major airports could be necessary to manage the company's debts. According to Ferrovial's finance director, Nicolas Villen, "We are working on the refinancing, we think it's possible but we have to study alternatives."