You can only apply for Affordable Care Act / Obama Care during the open enrollment period. For 2013-2014 year, the open enrollment is from October 1, 2013 to March 31, 2014. Afterwards, the open enrollment is from Oct 15 to December 7.
If you experience a qualifying event, then you can apply for health insurance during the special enrollment. Special enrollment period is usually 60 days from your qualifying event.
What is a Qualifying Event?
You become unemployed/layoff or you experience reduction in the number hours of work
Change in income
Change in Marriage Status (Marriage / Divorce)
Change in Dependents (Birth of new baby, adoption)
You moved to a new state that has different insurance coverage