Disability insurance covers a portion of your income in case you become disabled and cannot perform your job. For example, if you are a surgeon and your hand gets cut off in an accident, you are no longer able to perform your primary occupation as a surgeon. Disability insurance would pay you a portion of your income, maybe 60% or 65%, depending on the terms of the policy.
Employers often make disability insurance available to employees in the form of sick leave, short-term disability, and long-term disability. If your employer offers this benefit, start with that coverage and then look to an outside policy if you need more.
Long-term care insurance covers services and nursing home care needed with an extended illnesses. Suppose you have saved for years and are finally able to retire, only to have a sudden stroke. Your life savings may be depleted if you need nursing home care. Your spouse may end up with nothing unless you have long-term care insurance to help cover the costs of your medical and care needs.
This type of insurance is becoming more expensive, and several companies have recently stopped offering long-term care policies. It may become more difficult to purchase as the market changes and shrinks, but many people don't need to purchase this type of insurance until they are around 60.
So, disability insurance protects you now as it replaces earnings in case of injury, and long-term care insurance protects you later so you can keep your savings in case of extended illness.