For starters, you will need health insurance. Everyone needs medical care. Even if you're in good health and you don't get sick a lot, there is no way to anticipate accidents and injuries, and having regular checkups is a crucial part of living a long and healthy life. The cost of medical care is now so high, that one minor accident can send you into debt for years or bankrupt your family.
You might also need life insurance, though not everyone does. For example, if you're healthy and have no dependants, you probably don't need life insurance. But if you have a child, a spouse, a parent or sibling that depends on you financially, you'll want to make sure they're taken care of should anything happen to you. Even if you don't have dependents, you might just want a policy that will at least relieve your relatives of the burden of funeral costs.
Now, here's where it gets really morbid - what if you're in an accident and you don't die, but become incapacitated and unable to care for yourself? That can be a huge burden on your family, and even more expensive than funeral costs. That's where long-term care insurance comes in. And if you're lucky enough to live a long, healthy life, long-term care insurance will cover nursing home costs for you when you're elderly, or even pay for your elderly parents' stay.