The health care system was structured poorly before any of this got started. Basically, how health insurance worked in the US is that about 90% of people get health insurance through their employer. This happened for two reasons: during the wage freezes around World War II, companies started offering fringe benefits to get around it, and there is a standard tax deduction for companies to offer health insurance. Individuals cannot claim this deduction.
For the remaining 10%, about half of them buy insurance on the open market, which is really cheap and easy for people that are young and healthy, expensive for older people, and pretty much impossible if you have an existing condition.
The remaining 5% go uninsured either by choice or because they can't afford insurance at all. That doesn't mean they can't get health care, that just means they use free clinics and pay cash at places like urgent care centers.
The ACA made changes to the insurance market. Among the changes were new minimum requirements for what companies could sell as "health insurance". For example, even if you are a single 25-year-old male, you now have to pay for maternity coverage, mental health coverage, childhood mental health care, and other things that you're very unlikely to need. Further, health insurance is no longer allowed to have a cap on coverage, and companies are not allowed to charge old, sick people more than a small multiple of what they charge young and healthy people.
Mind you, a key campaign promise while passing the bill was that "if you like your health insurance you can keep it".
After the bill was passed in rules were promulgated, it turns out that if health insurance companies make any change at all in a plan (even changing the deductible by five dollars), then they are no longer able to offer the old plan and have to switch to a plan that qualifies.
This rule is resulting in millions of people having their health insurance canceled, and they are being forced to go on to the (nonworking) exchanges to buy health insurance that in most cases is much more expensive (if you're relatively healthy and young). New insurance results in you having to pick all new doctors, most likely, also.
So the main opposition comes from conservatives who oppose it on the grounds that government is interfering in the healthcare market, healthy people who are now being forced to pay sometimes more than twice as much, and young people who now get to subsidize the insurance of old people.
An additional philosophical objection comes from the individual mandate, which technically was upheld by the Supreme Court but many conservatives think that the Chief Justice had to contort himself into a pretzel to do it. Essentially, the ruling said that the government doesn't have the power to force you to buy something but they do have the ability, under the taxing power to tax you for not buying something. This was not written into the law, and the Supreme Court had to, in effect, change a clause in the law to make this work.
There is a populist opposition to the law because there are many large companies and labor unions (and Congress itself, including the president) that have been exempted from the law... And there are rumors that the president is going to further exempt certain well-connected labor unions from the tax on high-cost health plans.
There are legal objections to the law because the president decided to postpone a tax on employers that was written into the law. Many legal conservatives don't believe that the president has the power to rewrite portions of a law after it's been passed by Congress.
There are religious objections the law because it requires private companies run by very religious people to offer abortion and contraception coverage in the health insurance they offer their employees.
And finally, because the Democrats and the President decided to pass this law without any Republican votes (And using a questionable tactic called reconciliation that enabled them to pass it with 50 votes instead of 60 in the Senate), there is no political downside to opposing the law on the Republican side. Mostly, Republicans dislike the law and therefore their representatives in Congress get more kudos as their objections to the law get louder and louder.
And there is no end in sight.