As a congressional disagreement over the Affordable Care Act caused the government to shut down early Tuesday morning, the health insurance marketplace mandated by the law opened for business.

The law, better known as Obamacare, created insurance exchanges where individuals and families can shop for insurance coverage. By law, the health care plans on offer cannot deny people coverage due to pre-existing conditions. In addition, they cover a list of "essential health benefits," such as prescription drugs, lab services and maternity care.

The rollout of the exchanges comes after a long and tortuous path for the Affordable Care Act. Since the law's passage in 2010, the House of Representatives has voted more than 40 times to repeal or block the act in other ways.

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In addition, much of the act survived a Supreme Court challenge, in a high-profile June ruling. Obamacare is also currently a sticking point in Congress' ongoing fiscal fights. House Republicans on Sunday voted to continue funding the government in exchange for a one-year delay in the act's implementation, a bill the Senate voted down, creating a government shutdown that started at midnight Tuesday.

In a Tuesday afternoon briefing, President Obama referred to the bill's staying power, despite its many challenges over three years.

"This is a law that passed both houses of Congress, a law that bears my signature, a law that the Supreme Court upheld as constitutional, a law that voters chose not to repeal last November," he said.

Amid the sound and fury over the broader law, the exchanges will open quietly, awaiting customers in need of insurance. As with any new initiative, there will likely be some speedbumps, says one expert.

[READ: 10 Effects of a Federal Government Shutdown]

"It's got a lot of moving parts to it, and in things with a lot of moving parts, something usually goes wrong initially," says James Brasfield, professor of management at Webster University in St. Louis. He says the program may largely run smoothly but there will likely also be some complications as private insurers and the government work together to coordinate information. In addition, there is the possibility for technical glitches in the exchange websites, Brasfield says.

Consumers will have help in making their choices, however, in navigators – people and organizations trained to assist exchange users in making insurance choices.

"Navigators are going to be key," says Kate Bicego, consumer program assistance manager at Healthcare for All Massachusetts, a nonprofit organization that promotes healthcare access. "They spread across statewide and helped people, helped to educate people about the law and helped to enroll and keep their coverage. ... They were pivotal to the success of healthcare reform."

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Customers will be able to purchase insurance starting Tuesday, but the coverage will not go into effect until Jan. 1, 2014. That means customers don't have to start shopping immediately on Oct. 1. And in fact, Brasfield says, it might be best if everyone didn't jump in all at once. A gradual entry of consumers could make it easier for administrators to work out any glitches.