My rough understanding is that it is insurance you pay just in case you are sued by a patient for making an error. So do you have to worry about this as a resident?
The hospital or university program that employs you will take care of the malpractice insurance during residency. When you go into private practice, you or your medical group will be responsible for purchasing your own insurance. It can range in price from as little as $7,000 a year to over $100,000, depending on your specialty, and the malpractice law of the state you practice in.
What is academic vs private?
Academic means you are employed by a medical school or university hospital, and they pay your salary and provide you with an office, and will typically pay for your expenses. You will usually share office space with other employed physicians. The salary structure is often quite complex. There is often a base salary, and a formula which will allow you to earn more money, but my understanding is that it's quite difficult to earn more than the base due to how the salaries are structured.
Private practice is in distinction to being employed. You can be employed by a university, in which case you are academic, or you can be employed by a hospital, or by a government agency, or another entity ( eg VA Hospital, US military, a school, a prison, a government health agency, etc ).
Private practice can mean that you have your own completely private practice, alone or with partner(s). You get paid by patients, either paying cash or by accepting insurance. You get paid based on each patient visit or each procedure you perform.
You can also work for a large HMO or large medical group, which can technically still be private practice, if you are in a large medical group, but practically speaking you are functioning as an employed physician, earning a flat salary, more or less.
And do most doctors face malpractice lawsuits in their career at least once?
Yes
Does it hurt your career as a physician permanently if you face a malpractice lawsuit?
Not usually, but it depends on what happened, if you actually did something wrong, what it was, and how often you get sued. Also, on the size of the community you practice in.
The shortened amount of time is detrimental to the training.
I have heard people say this. Residents don't like the handoffs. However, very few people experienced residencies both ways, so as to be able to compare them. As someone who did a surgery residency before the 80 hour rule, I can tell you that it was no fun working those long hours. My residency schedule had long weeks and short weeks every month. The short week was about 80 hours. The long week was between 130 and 140 hours. The long weeks were not easy, and probably were not very educational. I would have loved an 80 week limit, despite the fact that now I get to talk about how hard it was back in the "days of the giants".