Final answer:
Barack Obama signed the Patient Protection and Affordable Care Act, also known as Obamacare, which includes an individual mandate requiring all Americans to purchase health insurance or face a penalty.
Step-by-step explanation:
The type of health insurance option that Barack Obama signed into law is the Patient Protection and Affordable Care Act (ACA), commonly known as Obamacare. The ACA significantly reformed the American healthcare system, with the intention of extending health insurance coverage to more Americans. It includes an individual mandate which requires that every individual buy insurance or face a penalty, providing various private insurance plans through state marketplaces, and includes subsidies to help make insurance more affordable based on income levels.