Medicare is a name given to government healthcare programs in Australia, Canada, and the United States. As the United States is the only country that matters, we will ignore the other two. In the U.S., Medicare is a quasi-universal health care program for certain sectors of the population. The modern program was proposed by President Lyndon Johnson as part of his "Great Society" programs. Medicare provides health insurance for persons 65 and older, as well as the long-term disabled and persons with end-stage renal disease and amyotrophic lateral sclerosis (ALS, also known as "Lou Gehrig's disease"). In 1965, when Medicare was enacted, only 65% of U.S. seniors had any kind of health insurance, and it was usually three times as expensive as that of other citizens.
As it is an attempt to use the government to help people, conservatives have opposed it from day one. Ronald Reagan released a record opposing "socialized medicine" and a particular bill that was a precursor to Medicare:
(I wonder if he'd bothered asking anyone in Western Europe if they were no longer free?)
Johnson was sympathetic to the idea of universal health care, but in the face of such opposition, decided it would be more politically feasible to push for limited programs (Medicare and Medicaid) that only covered certain segments of society, such as the elderly, some disabled people, and children of poor families. It's hard for politicians to stand up to emotional appeals like "Do you want Grandma to die because she can't afford health care?", whereas they can argue that younger adults bankrupted by medical bills should have "worked harder" and "been more responsible". Programs like Medicare also have the, er, "benefit" of taking sicker and more expensive patients off the backs of private insurers, while younger and healthier people must still turn to them for coverage, effectively subsidizing insurance companies.
Medicare is a "social insurance" program. Anyone belonging to one of the eligibility categories can enroll. The Medicare program today has four "parts". Originally there were just Parts A and B; these are today sometimes referred to as "Original Medicare".
 Part A
Part A is hospital and hospice insurance. It is mostly funded by a national payroll tax. Once a person, or their spouse (or former spouse in some cases), has paid this tax for 10 years, they can enroll in Part A for free when eligible. Otherwise, there is a monthly premium.
 Part B
Part B is "medical insurance". Today this is basically anything that doesn't fall under the other parts. This includes things like outpatient doctor's visits and durable medical equipment. Part B is "optional", in the sense that one can enroll only in Part A if they desire (but not the inverse; if you enroll in Part B you must also enroll in Part A). Those who choose to enroll in Part B pay a monthly premium (on top of any Part A premium). If you're wondering why this is separate, it's because health care costs, particularly for drugs and outpatient procedures, were a lot cheaper in the 1960s when Medicare was designed. And at the time some people had cushy healthcare benefits from a current or former job, at least for outpatient stuff (like so-called "fee-for-service" plans). The idea was to extend coverage to everyone for the things likely to be expensive, like hospital stays, while giving people the option of paying for additional coverage. Today relatively few people only enroll in Part A, since most people eligible for Medicare have no other health coverage (most private health insurance policies today automatically drop you if you become eligible for Medicare). A lot of people enrolled in Part B forget that they pay an additional premium, because if you receive Social Security benefits (which most people in Medicare do) you can choose to have the premium automatically deducted from your benefit.
 Part C
Part C comprises "Medicare Advantage" plans. Experiments with these plans began in the 1980s. Anyone enrolled in both Parts A and B has the option of enrolling in a Medicare Advantage plan to receive their Medicare benefits. These are plans administered by private insurance companies. Medicare pays them a per-capita fee for each enrollee. Typically, enrollees must receive all of their covered care through the plan, which usually is an HMO, meaning they can only use "in-plan" providers if they want to be covered. Despite this, these plans are often attractive, as they frequently offer lower out-of-pocket costs than Original Medicare. The plans are regulated, and no plan is allowed to offer worse benefits than Original Medicare. Some offer added benefits, like dental insurance, for an additional premium. Enrollees can switch between Original Medicare and Part C, or between different Part C plans, during an annual open enrollment period, as well as under "special enrollment periods" under certain conditions.
Yes, that's right, private insurance companies participate in Medicare, and even make money off of it. Tyrannical socialism at work, folks!
 Part D
Part D is prescription drug insurance. Medicare originally only covered medications administered as part of an inpatient stay, under Part A. This was not a big deal in the '60s as drug costs were lower, and there were fewer people taking drugs regularly. Medicare beneficiaries had the option of buying private "add-on" insurance that covered prescription drugs. However, as life expectancy and drug costs went up over the decades, drug costs became a huge burden. Elderly people taking trips to Canada or Mexico to buy cheaper drugs became common. George W. Bush made adding prescription drug coverage to Medicare a prominent item on his 2000 campaign agenda, and managed to get enough Republicans in Congress to go along (this was pre-Tea Party). Part D went into effect in 2006. It is offered through private insurers, either as standalone Part D plans for those on Original Medicare, or as part of Part C Medicare Advantage plans.
One big issue with Part D is that it involves one of the most naked corporate handouts in recent U.S. history. As part of the law establishing Part D, Medicare was forbidden from negotiating drug prices with manufacturers. As a consequence, Medicare pays substantially more for drugs than it ought to. Private insurers negotiate prices all the time, and it is likely that Medicare, given its huge coverage base, could negotiate significant discounts. Even more hilariously, the Department of Veterans Affairs is allowed to negotiate drug prices, and (surprise!) they pay lower costs than Medicare, despite having fewer patients. Republicans have resisted all calls to remove this prohibition, because socialism or something. So the next time someone calls themselves a "fiscal conservative", ask them if they support allowing Medicare to negotiate drug prices. If they say no, you know that their profession of being a "fiscal conservative" is bullshit: usually coded language either for "don't give welfare to those darkies", or for an agenda of "starving the beast" (allow Medicare costs to keep going up unnecessarily, and then argue that Medicare is just too expensive and we have to "reform" it; see below).
 Medicare and Medicaid
Medicaid can be considered a kind of "sister" program to Medicare. Both were part of Johnson's "Great Society" programs, and both are administered by the same U.S. government department, the Centers for Medicare & Medicaid Services. Medicaid is a means-tested welfare program that offers limited health coverage to the poor. It is possible to qualify for both Medicare and Medicaid. Such people are often referred to as "dual-eligible". If one is eligible for both, they "stack": Medicaid will pay some or all of a beneficiary's Medicare premiums, out-of-pocket costs, etc. This means that, with Medicare now including prescription drug coverage, Medicare and Medicaid together form a de facto, inefficient, universal health care program for the elderly and severely disabled.
 "Saving" Medicare by destroying it
Medicare currently covers about 15% of the US population (48 million people), mostly senior citizens. Many conservatives recognize that it's impossible to overtly attack Medicare, as the near entirety of their voting base relies on it and it is quite popular, so instead they talk about "strengthening" Medicare.
The ongoing retirement of the "Baby Boom" generation along with the high level of healthcare inflation seen in recent years mean the ongoing viability of Medicare is a concern. However, the Affordable Care Act has led to a flattening of healthcare inflation, leading to this no longer being as much of an issue. This has not stopped conservatives from trying to "fix" Medicare by limiting benefits and excluding beneficiaries.
Typical of this approach was the Paul Ryan budget, which would have completely replaced Medicare with fixed-value vouchers to buy private health insurance for new enrollees starting in 2022. These vouchers would have been indexed to overall inflation, not healthcare inflation, leading to a gradual "stealth" reduction in real benefits if healthcare costs continued to outpace inflation. It also would have gradually increased the enrollment age from 65 to 67, despite the fact that increasing eligibility age would do nearly nothing to cut costs.
One exception to this pattern was George W. Bush, who really did strengthen Medicare by pushing for the addition of pharmaceutical drug coverage ("Part D"). Hard-line "fiscal conservatives" point to this as one of the ways in which Bush was not a "true conservative."
Despite most claims to the contrary, Medicare has proven repeatedly to be one of the least wasteful parts of the U.S.'s overall health care system, providing better and cheaper coverage than most private insurance. During the run-up to the Affordable Care Act, many liberals proposed extending universal health care through "Medicare for All," expanding the program to cover the entire U.S. population. While utterly sensible in every way, it was a non-starter politically.
The overall success of Medicare has led many of those who use it to forget that it is a government program. This led to hilarious signs held by those who opposed the Affordable Care Act demanding the government stay out of Medicare.