Here is an interesting post from the Buck Institute on sources of funding for fundamental research into aging, with tables listing the various contributing organizations. While looking through the list, it is worth bearing in mind that for really early stage, high risk, novel research the largest sources are unavailable. NIA grants, for example, only become a possibility once you've actually made the initial breakthrough and have early proof that you have achieved something new. This is a systematic issue in medical research, and it is why philanthropic donations are essential for progress. Few really important novel attempts to advance the state of the art are directly funded at the outset by large institutional sources like the NIA or large pharmaceutical companies, though there is certainly a lot of creative bookkeeping that takes place in larger laboratories in order to split off the necessary funds for early stage, prospective work. Without that very early stage work there would be no progress, but most funding sources - public and private - act as though the prototypes they are willing to fund come into existence from nothing, as if by magic.
Where does the money to fund basic aging research come from? After all, scientists need to be paid, purchase supplies for their research, and somehow find the money to attend conferences to talk about their results. In the US at least (the funding situation is different in the UK), money comes primarily in the form of grants from the federal government, which both pay the salaries of researchers and provide them with money for their experiments.
The National Institute of Health (NIH, a federal agency) is huge and awesome. The main mechanism by which money it distributes money is through "R" grants. These are large (~$1 million), multi-year grants awarded to principal investigators (usually professors) at research institutions who go through a competitive process to apply for them. About 90% of non-profit aging research project funding comes from the NIH, and most of that is in the form of "R" grants. NIH funding has shrunk in real terms by 11% since 2003. Thankfully the NIA, the wing of the NIH, is one of the few institutes who have seen extra budgetary support in recent years.
Apart from the NIH, there are several private foundations that support aging research and specific diseases of aging. Budgets are from the latest available information, and frankly I was surprised by how small this chunk is. Don't get me wrong, each of these foundations are great and their funds support promising scientific projects and programs. But all together, they're less than 10% of the annual R-grant budget (note that a different situation exists in the UK, where the giant Wellcome Trust funds about $600 million in biomedical research). A lot of private giving to aging research is not structured as annual grant programs, though. For example, at the Buck we receive generous one-time donations from local businesses, individuals, and some of the aging foundations listed below to support our facilities.
There are also a bunch of institutes and research departments dedicated to basic aging research. A lot of universities and medical schools have some department with "aging" or "gerontology" or "geriatrics" in their name. Each of these typically distributes intramural funds. Want their money? Get a job there.
But if we move outside academic research to money spent on commercialized research applications by private companies, the pie changes quite a bit. In aggregate, drug companies outspend the NIH on R&D every year by over $20 billion. The precise portion of this going towards "aging research" is hard to measure. While most aging research at drug companies is not focused on aging itself, diseases of aging such as diabetes, heart disease, and cancer are intense areas of study. Recent years have seen the founding of private companies dedicated specifically to aging research. It is hard to guess at annual budgets for these new players, but they're pretty huge. Calico's $500 million in committed funds, for example, is over half the amount the NIA spends on R grants in a year.