It may sound contradictory, but according to scientist and statistician Hans Rosling, the main criterion to limit population growth is a reduction of the child mortality rate (see also his Ted Talk on the topic and the stats below).
Child mortality rate vs Country population growth rate (including migration) in 2011,
Each bubble is a country, bubble color indicates continent, bubble size relates to number of babies per woman
In the next decades most population growth is expected to be in developing countries where child mortality is high. This is especially the case in the sub-Saharan countries, where children are an insurance that there is someone to take care of you when you are old. Population growth rates in developed countries are much lower. In some countries like Germany, Russia and Japan it is even zero or negative (not counting recent immigration, see also wikipedia)
Lower child mortality can be achieved by better living conditions and better health services. This in turn can be achieved by better education and higher income. Also women who are better educated tend to marry later and get children at a higher age.
As Móż has also mentioned in his comment, another big influence on population growth can be government policy and incentives. China for example still has one of the lowest population growth rates in Asia (although I do not consider their former 1-child policy ethical).
More information about population growth can be found on the website of Hans Rosling's Gapminder.org organisation. The site also has a great interactive graphical statistics tool that shows the relationship between child mortality, babies born per woman, education, income, health services and many other factors, plus their changes since 1960.