Anyone age 65 or over is eligible for Medicare. Most people age 65 and over are covered under Medicare Part A for free, based on their work records or on their spouse's work records.
People over 65 who are not eligible for free Medicare Part A coverage can enroll in it and pay a monthly fee for the same coverage. The premium base rate depends on the number of work credits you've earned. If you pay for Part A hospital insurance, you must also enroll in Part B medical insurance, for which you pay an additional monthly premium. Note that the Medicare Part A premium increases by 10% for each year after your 65th birthday that you wait to enroll.