It covers you if you die for the face amount of the policy just like any other form of life insurance as long as you die within the term of the policy. It doesn't matter what you die from as long as you do not lie on the application or commit suicide the first two year of the policy.
Term life insurance covers death by any cause except suicide in the first two years. This includes death by illness (even if due to a pre-existing condition), death caused by accident (including while flying or participating in dangerous hobbies), and death by unknown causes (such as old age).
If your life insurance application asked about these things and you disclosed them, you are covered. If, a few years after taking out the Term life insurance policy, you developed a health condition or took up a dangerous hobby that led to death, that is covered, too. Of course, the policy must be in effect at the time of death, which for Term insurance means that you have paid regular premiums and not allowed the policy to lapse.
Term life insurance generally covers a temporary economic liability for the beneficiaries of the insured. It is highly recommended that you purchase a term life insurance policy that includes the right to convert the policy to a permanent life insurance policy in case the temporary liability becomes permanent. The duration of your economic liability should dictate the length of guaranteed coverage you’re applying for. The death proceeds, in theory, should fund the liabilities for the surviving beneficiaries, so they’re not burdened bills at your demise.
Term life insurance covers you and your family's (i.e. your spouse and children) lives. If you are speaking in terms of illnesses or health issues, then that would depend on the company. Each company has different guidelines and risk classes in reference to what they will accept and will not accept. The amount of coverage is based on your temporary financial needs. If you die from anything (other than suicide, as mentioned above) your family will be compensated to financially accommodate their loss. Unfortunately the insurance company can not emotionally accommodate your family's loss; but at least they won't suffer financial burden as long as you have the right coverage in place.
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If your life insurance application asked about these things and you disclosed them, you are covered. If, a few years after taking out the Term life insurance policy, you developed a health condition or took up a dangerous hobby that led to death, that is covered, too. Of course, the policy must be in effect at the time of death, which for Term insurance means that you have paid regular premiums and not allowed the policy to lapse.