On a very basic level, there are three main ways in which people pay for health care. First, there is a purely private system, in which individuals pay out of their own pockets if they can afford it. For many underdeveloped countries, this is still the norm. Another method of paying for health care is to consider it a basic human right (as it is enshrined in the Canadian Charter of Rights and Freedoms, for example), and pay for it out of public funds, as is the case in most OECD (rich) countries except the United States. A third source is "pooled risk", in which either public or private insurance companies pool the funding (whether public or private), taking in money per capita and expending it as needed. These third party insurers may be either private citizens or a public single payer.
Although these three systems are completely independent of the age of the populace, forms of pooled risk or public benefit payment run into demographic difficulties of many people (impoverished elderly, children, non-working adults, students) are getting care without making contributions (whether in the form of paying taxes or contributing to for-profit insurance plans). For-profit systems, such as the US insurance companies, also carry the additional burden that sick people are paying not only for health care but for the multi-million dollar salaries of executives and shareholder profits.